Chapter 1 Problems

Thirty chapter problems: 10 from the original book and 20 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem

   Calculate ${\left(\text{csc}\left(\frac{5\pi }{4}\right)•\text{tan}\left(\frac{5\pi }{4}\right)\right)}^2-1$. Refer to the unit circle.

   Show your work for Problem 1

   

   Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

   Your written working

   Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

   Written working stays in this browser.

   Worked solution to problem 1 →

2. Original problem

   Calculate $\frac{\text{cot}\left(\frac{\pi }{6}\right)\text{csc}\left(\frac{\pi }{6}\right)}{\left(\text{sec}\left(\frac{\pi }{8}\right)+\text{tan}\left(\frac{\pi }{8}\right)\right)•\left(\text{sec}\left(\frac{\pi }{8}\right)-\text{tan}\left(\frac{\pi }{8}\right)\right)}$

   Show your work for Problem 2

   

   Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

   Your written working

   Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

   Written working stays in this browser.

   Worked solution to problem 2 →

3. Original problem

   Calculate ${\left(\text{csc}\left(\frac{3\pi }{4}\right)•\text{tan}\left(\frac{3\pi }{4}\right)\right)}^2+{\left(\text{sec}\left(\frac{3\pi }{4}\right)•\text{cot}\left(\frac{3\pi }{4}\right)\right)}^2$

   Show your work for Problem 3

   

   Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

   Your written working

   Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

   Written working stays in this browser.

   Worked solution to problem 3 →

4. Original problem

   Write the equation for the graph below with sin(x) as the base function (Hint: Notice the maxima and minima).

   Show your work for Problem 4

   Source sinusoidal graph

   A blue sinusoidal curve is drawn on a Cartesian grid. The horizontal axis is x and the vertical axis is y. Visible labeled x-values include −7.5, −5, −2.5, 0, 2.5, 5 and 7.5; visible labeled y-values include −2.5, 0, 2.5 and 5.

   The curve has a trough on the y-axis at approximately y=−2 and neighboring crests at approximately y=4, near x=−6.3 and x=6.3. It crosses the x-axis near x=−2.5 and x=2.5. The plotted wave is symmetric about the y-axis.

   Coordinates not printed next to the extrema are approximate readings from the source raster. No exact symbolic crest positions or period are printed; this description does not supply an unprinted exact period or an equation.

   

   Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

   Your written working

   Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

   Written working stays in this browser.

   Worked solution to problem 4 →

5. Original problem

   Write the equation for the graph in number 4 with cos(x) as the base function rather than sin(x).

   Show your work for Problem 5

   Source sinusoidal graph

   A blue sinusoidal curve is drawn on a Cartesian grid. The horizontal axis is x and the vertical axis is y. Visible labeled x-values include −7.5, −5, −2.5, 0, 2.5, 5 and 7.5; visible labeled y-values include −2.5, 0, 2.5 and 5.

   The curve has a trough on the y-axis at approximately y=−2 and neighboring crests at approximately y=4, near x=−6.3 and x=6.3. It crosses the x-axis near x=−2.5 and x=2.5. The plotted wave is symmetric about the y-axis.

   Coordinates not printed next to the extrema are approximate readings from the source raster. No exact symbolic crest positions or period are printed; this description does not supply an unprinted exact period or an equation.

   

   Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

   Your written working

   Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

   Written working stays in this browser.

   Worked solution to problem 5 →

6. Original problem · clarified

   If the function f(x) = $e^x$ were shifted one unit to the right, one unit vertically up, and vertically compressed by $\frac{1}{2}$ , what would be the new function?

   Clarification: Apply the vertical compression before the upward translation; equivalently use the combined transform g(x)=a f(x−h)+k.

   Show your work for Problem 6

   

   Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

   Your written working

   Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

   Written working stays in this browser.

   Worked solution to problem 6 →

7. Original problem

   What is the domain and range of the function f(x) = ${\text{sin}}^{-1}(x)$ ? Refer to the unit circle.

   Show your work for Problem 7

   

   Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

   Your written working

   Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

   Written working stays in this browser.

   Worked solution to problem 7 →

8. Original problem

   Convert the equation $y={\text{log}}_4(20)+{\text{log}}_{10}(16)$ into the form y = ln(c). Then find the value of y. You may want to refer to log rules.

   Show your work for Problem 8

   

   Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

   Your written working

   Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

   Written working stays in this browser.

   Worked solution to problem 8 →

9. Original problem

   Simplify $y=\text{ln}{\left(\frac{3e}{\pi }\right)}^2$ such that the argument of the ln function only has one term (ex: $y=\text{A ln}(a)+\text{B ln}(b)+\text{C ln}(c)$. Refer to the log rules

   Show your work for Problem 9

   

   Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

   Your written working

   Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

   Written working stays in this browser.

   Worked solution to problem 9 →

10. Original problem · clarified

    (You knew this one was coming) Given that the black graph is y = $e^x$ , write the equations of all of the other colored graphs.

    Clarification: Identify the standard exponential/logarithmic reflections that match the labeled raster graph. The equations are inferred from the graph.

    Show your work for Problem 10

    Source colored-curve graph

    A Cartesian plot shows the black reference curve named in the problem, y=e^x, together with seven colored curves. The displayed x-axis is labeled from −4 to 4 and the y-axis includes labels from −3 to 3. The black curve increases through (0,1).

    The green curve is positive, decreases through (0,1), and approaches the x-axis toward the right. The blue curve is negative, decreases through (0,−1), and approaches the x-axis toward the left. The orange curve is negative, increases through (0,−1), and approaches the x-axis toward the right.

    The pink curve is drawn to the left of the y-axis, increases through (−1,0), and becomes steep upward near that axis. The cyan curve is also drawn to the left of the y-axis, decreases through (−1,0), and becomes steep downward near that axis.

    The purple curve is drawn to the right of the y-axis, decreases through (1,0), and becomes steep upward near that axis. The yellow curve is drawn to the right of the y-axis, increases through (1,0), and becomes steep downward near that axis.

    Color, side of the axis, plotted intercepts and visible direction describe the raster. No equations for the seven colored curves or numerical values beyond its labeled grid/intercepts are added.

    

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 10 →

11. Added problem

    Quick unit-circle check: find $\sin (11\pi /6)·\sec (11\pi /6)$ exactly.

    Show your work for Problem 11

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 11 →

12. Added problem

    Simplify $\tan \theta ·\cot \theta$, and keep the original domain restrictions.

    Show your work for Problem 12

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 12 →

13. Added problem

    Solve $2\sin x=1$ for $0\le x<2\pi$.

    Show your work for Problem 13

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 13 →

14. Added problem

    If $\sin \theta =3/5$ and θ is in quadrant II, find $\sin \left(2\theta \right)$.

    Show your work for Problem 14

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 14 →

15. Added problem

    List the amplitude, period, midline, and horizontal shift of $y=-2\cos (3x-\pi /2)-4$.

    Show your work for Problem 15

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 15 →

16. Added problem

    A sinusoid has maximum 3, minimum −5, and successive peaks at $x=0$ and $x=6$. Write one equation.

    Show your work for Problem 16

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 16 →

17. Added problem

    Move $\sqrt{x}$ left 3, reflect across the x-axis, stretch vertically by 2, then move up 4. Give the equation, domain, and range.

    Show your work for Problem 17

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 17 →

18. Added problem

    A table gives $f(-1)=2$, $f\left(0\right)=4$, and $f\left(1\right)=1$. If $g\left(x\right)=3f(x+1)-2$, find $g(-1)$ and $g\left(0\right)$.

    Show your work for Problem 18

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 18 →

19. Added problem

    Stretch $e^x$ vertically by 3, shift it right 2 and down 1. Give the function, its horizontal asymptote, and its value at $x=2$.

    Show your work for Problem 19

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 19 →

20. Added problem

    Expand $\ln \left(x^2\right(x+1)/5)$ assuming $x>0$.

    Show your work for Problem 20

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 20 →

21. Added problem

    Combine $2\ln x-\ln (x-1)+\ln 5$ into one log. State the original domain.

    Show your work for Problem 21

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 21 →

22. Added problem

    Solve $\ln (x-1)+\ln (x+1)=\ln 8$ over the real numbers.

    Show your work for Problem 22

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 22 →

23. Added problem

    Evaluate ${\log }_{4}8$ exactly using change of base.

    Show your work for Problem 23

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 23 →

24. Added problem

    For $f\left(x\right)=x^2+1$ restricted to $x\ge 0$, find its inverse and the inverse domain.

    Show your work for Problem 24

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 24 →

25. Added problem

    Compare $\arcsin \left(\sin \right(5\pi /6\left)\right)$ and $\arccos \left(\cos \right(5\pi /6\left)\right)$. Explain the difference.

    Show your work for Problem 25

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 25 →

26. Added problem

    Let $f\left(x\right)=\ln x$ and $g\left(x\right)=e^x$. Find both compositions and their domains.

    Show your work for Problem 26

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 26 →

27. Added problem

    For $y=\tan \left(2x\right)-1$, state the period and every excluded input.

    Show your work for Problem 27

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 27 →

28. Added problem

    You know $f\left(0\right)=1$ and $f\left(1\right)=3$. A transformed curve $g\left(x\right)=af(x-2)+b$ passes through $(2,3)$ and $(3,-1)$. Find a and b.

    Show your work for Problem 28

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 28 →

29. Added problem

    A friend claims $1+{\tan }^2\theta =\sec \theta$. Find the mistake and give a counterexample.

    Show your work for Problem 29

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 29 →

30. Added problem

    Find the inverse of $g\left(x\right)=2e^{x-3}+5$ and state its domain.

    Show your work for Problem 30

    Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.

    Your written working

    Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.

    Written working stays in this browser.

    Worked solution to problem 30 →