Functions and Inverse Functions Worksheet

**Page 1**

1.

Which of the following is the graph of a relation, that has an inverse, which is a function?

a. | Graph B | ||

b. | Graph C | ||

c. | Graph A | ||

d. | None of these |

[Horizontal line test.]

For the choice A: Since, the graph intersects the horizontal line at more than one point, hence it fails in the horizontal line test.

For the choice B: Since, the graph intersects the horizontal line at more than one point, hence it fails in horizontal line test.

For the choice C: Since, the graph intersects the horizontal line at one point, hence it passes through horizontal line test.

So, graph -C represents the relation that has an inverse, which is a function.

Correct answer : (2)

2.

Choose the graph of the inverse function for the graph shown.

a. | Graph-A | ||

b. | Graph-B | ||

c. | Graph-C | ||

d. | None of the above |

[The inverse reflection principle.]

Take the given graph and choose a point (2, 1.25).

[Read the value of

Here we choose (2, 1.25) arbitrarily on the given graph of the function then the point (1.25, 2) must lie on the graph of itÃ¢â‚¬â„¢s inverse function.

For the graph - A:

At

Hence, it is not the graph of the inverse function of the given graph.

[From the graph - A.]

For the graph - B:

At

Hence, it is not the graph of the inverse function of the given graph.

[From the graph - B.]

For the graph - C:

At

Hence, it is the graph of the inverse function of the given graph.

[From the graph - C.]

Correct answer : (3)

3.

Which of the ordered pairs are in the relation 3$x$ + 4$y$ = 8?

a. | (1, 1) | ||

b. | (3, - 2) | ||

c. | (4, - 1) | ||

d. | (- 2, 3) |

3(1) + 4(1) = 7 ≠ 8

For (1, 1),

[Substitute

3(3) + 4(- 2) = 1 ≠ 8

For (3, - 2),

[Substitute

3(4) + 4(- 1) = 8 = 8

For (4, - 1),

[Substitute

So, (4, - 1) is in the relation defined by 3

Correct answer : (3)

4.

If $f$ and $g$ are two functions, then the function ($f$ + $g$) is defined only when

a. | Domain of $f$ ≠ Domain of $g$ | ||

b. | Both $f$ and $g$ have intersecting domains | ||

c. | The new function ($f$ + $g$) has the domain of $g$ only | ||

d. | The new function ($f$ + $g$) has the domain of $f$ only |

Correct answer : (2)

5.

If $f$($x$) = $x$^{2} and $g$($x$) = $\sqrt{x+7}$, then find ($f$ + $g$).

a. | $x$ ^{2 } + $\sqrt{x-7}$ | ||

b. | $x$ ^{2 } - $\sqrt{x+7}$ | ||

c. | $x$ ^{2 } + $\sqrt{x+7}$ | ||

d. | $x$ + $\sqrt{{x}^{2}+7}$ |

(

[Sum rule of functions.]

=

Correct answer : (3)

6.

If two functions are defined by $f$($x$) = 6$x$ + 2 and $g$($x$) = $x$ + 9, then find $g$o$f$(2).

a. | 24 | ||

b. | 14 | ||

c. | 23 | ||

d. | 9 |

[Use the definition of composite function.]

=

[Substitute

=

= 14 + 9

[Substitute

= 23

Correct answer : (3)

7.

If two functions are defined by $f$($x$) = $x$^{2} - 12 and $g$($x$) = $\sqrt{x+6}$, then what is ($f$o$g$)($x$)?

a. | $x$ - 6 | ||

b. | $x$ + 6 | ||

c. | 6 - $x$ | ||

d. | $\sqrt{x}$ + 6 |

(

[Use the definition of composite function.]

=

[Substitute in

=

[Substitute in

=

Correct answer : (1)

8.

If two functions are defined by $f$($x$) = $x$^{2} - 3 and $g$($x$) = $\frac{1}{x-9}$, then write the domain of the composite function ($f$o$g$)($x$).

a. | R - {9} | ||

b. | (-1, ∞) | ||

c. | (0, ∞) | ||

d. | R |

(

[Use the definition of composite function.]

=

[Use

The function (

Correct answer : (1)

9.

If the functions are defined by $f$($x$) = $\sqrt{x+5}$ and $g$($x$) = | $x$ + 3 |, then what is the domain of the function $\frac{f}{g}$?

a. | [- 5, - 3] ∩ (- 3, ∞) | ||

b. | (- ∞, ∞) | ||

c. | (- 5, - 3) ∩ [ - 3, ∞) | ||

d. | [ - 5, - 3) $\cup $ (- 3, ∞) |

(

[Use the quotient rule of functions.]

=

[Use

The composite function is not defined when

Hence, the domain is [ - 5, - 3)

Correct answer : (4)

10.

Choose the direct relationship between $x$ and $y$ when $x$ = $t$^{6} and $y$ = $t$ - 35, where $t$ is a parameter.

a. | $y$ = ($x$ + 35) ^{6} | ||

b. | $x$ = ($y$ + 35) ^{6} | ||

c. | $x$ = ($y$ - 35) ^{6} | ||

d. | $y$ = ($x$ - 35) ^{6} |

[Write

[Substitute

Correct answer : (2)