Review the following concepts if needed:

- Graphs of Circular Functions Help
- Inverses of Circular Functions Help
- Graphs of Circular Inverses Help

**Graphs and Inverses Practice Test**

A good score is eight correct.

1. The sine function and the tangent function

(a) have identical shapes when graphed

(b) have different ranges

(c) have identical domains

(d) are inverses of each other

2. The restrictions on the domain and range of the inverse circular functions are necessary in order to ensure that:

(a) no negative angles are involved

(b) they never “blow up”

(c) none of them have more than one *y* value (ordinate) for any *x* value (abscissa)

(d) the domains are defined for all real numbers

3. The graph of the cosine function

(a) has the same shape as the graph of the sine function, but is “stretched” vertically

(b) has the same shape as the graph of the sine function, but is shifted horizontally

(c) has the same shape as the graph of the sine function, but is shifted vertically

(d) has the same shape as the graph of the sine function, but is “squashed” horizontally

4. The domain of the arccotangent function

(a) encompasses only the real numbers between, and including, –1 and 1

(b) encompasses only the values between 90° ( *π/2* rad) and 270° (3π/2 rad)

(c) encompasses only the real numbers less than –1 or greater than 1

(d) encompasses all of the real numbers

5. What does the expression (sin *x* ) ^{–1} denote?

(a) The reciprocal of the sine of *x*

(b) The sine of 1/x

(c) The arcsine of *x*

(d) None of the above

6. Look at Fig. 3-11. Consider the interval *S* of all values of *y* such that *y* is between, and including, –1 rad and 1 rad. Which of the following statements is true?

(a) *S* constitutes part of the domain of the function shown in the graph

(b) *S* constitutes part of the range of the function shown in the graph

(c) *S* constitutes all of the domain of the function shown in the graph

(d) *S* constitutes all of the range of the function shown in the graph

7. Look at Fig. 3-13. What can be said about this function based on its appearance in the graph?

(a) Its range is limited

(b) Its range spans the entire set of real numbers

(c) Its domain is limited

(d) It is not, in fact, a legitimate function

8. The graph of the function *y* = sin *x* “blows up” at

(a) all values of *x* that are multiples of 90°

(b) all values of *x* that are odd multiples of 90°

(c) all values of *x* that are even multiples of 90°

(d) no values of *x*

9. The function *y* = csc *x* is defined for

(a) only those values of *x* less than –1 or greater than 1

(b) only those values of *x* between, and including, –1 and 1

(c) all values of *x* except integral multiples of π rad

(d) all values of *x* except integral multiples of π/2 rad

10. Which of the following graphs does not “blow up” for any value of *x* ?

(a) The curve for *y* = arctan *x*

(b) The curve for *y* = tan *x*

(c) The curve for *y* = arccsc *x*

(d) The curve for *y* = csc *x*

**Answers:**

1. b

2. c

3. b

4. d

5. a

6. b

7. a

8. d

9. c

10. a

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