﻿ Functions and Inverse Functions Worksheet | Problems & Solutions

# Functions and Inverse Functions Worksheet

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

#### Solution:

The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original relation in at most one point.
[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.

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

#### Solution:

If we choose any point (a, b) arbitrarily on the given graph of the function then the point (b, a) must lie on the graph of its inverse function.
[The inverse reflection principle.]

Take the given graph and choose a point (2, 1.25).
[Read the value of y at x = 2 from the graph carefully.]

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 x = 1.25 the value of y is approximately 1.1. So, (1.25, 1.1) is the point on the graph.
Hence, it is not the graph of the inverse function of the given graph.
[From the graph - A.]

For the graph - B:
At x = 1.25 the value of y is - 2. So, (1.25, - 2) is the point on the graph.
Hence, it is not the graph of the inverse function of the given graph.
[From the graph - B.]

For the graph - C:
At x = 1.25 the value of y is 2. So, (1.25, 2) is the point on the graph.
Hence, it is the graph of the inverse function of the given graph.
[From the graph - C.]

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)

#### Solution:

3x + 4y = 8

3(1) + 4(1) = 7 ≠ 8
For (1, 1),
[Substitute x = 1, y = 1.]

3(3) + 4(- 2) = 1 ≠ 8
For (3, - 2),
[Substitute x = 3, y = - 2.]

3(4) + 4(- 1) = 8 = 8
For (4, - 1),
[Substitute x = 4, y = - 1.]

So, (4, - 1) is in the relation defined by 3x + 4y = 8.

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

#### Solution:

If f and g are two functions, then the function (f + g) is defined only when both f and g have intersecting domains.

5.
If $f$($x$) = $x$2 and $g$($x$) = , then find ($f$ + $g$).
 a. $x$2 + b. $x$2 - c. $x$2 + d. $x$ +

#### Solution:

f(x) = x2 and g(x) = x + 7

(f + g) (x) = f(x) + g(x)
[Sum rule of functions.]

= x2 + x + 7

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

#### Solution:

f(x) = 6x + 2 and g(x) = x + 9

gof(2) = g[f(2)]
[Use the definition of composite function.]

= g[6(2) + 2]
[Substitute x = 2 in f(x) = 6x + 2.]

= g[14]

= 14 + 9
[Substitute x = 14 in g(x) = x + 9.]

= 23

7.
If two functions are defined by $f$($x$) = $x$2 - 12 and $g$($x$) = , then what is ($f$o$g$)($x$)?
 a. $x$ - 6 b. $x$ + 6 c. 6 - $x$ d. $\sqrt{x}$ + 6

#### Solution:

f (x) = x2 - 12 and g (x) = x + 6

(fog)(x) = f(g(x))
[Use the definition of composite function.]

= f (x + 6)
[Substitute in g(x) = x + 6.]

= (x + 6)2 - 12
[Substitute in f(x) = x2 - 12.]

= x - 6

8.
If two functions are defined by $f$($x$) = $x$2 - 3 and $g$($x$) = , then write the domain of the composite function ($f$o$g$)($x$).
 a. R - {9} b. (-1, ∞) c. (0, ∞) d. R

#### Solution:

f(x) = x2 - 3 and g(x) = 1x - 9

(fog)(x) = f(g(x))
[Use the definition of composite function.]

= f(1x - 9) = 1(x - 9)2 - 3
[Use g(x) = 1x - 9, f(x) = x2 - 3.]

The function (fog)(x) is defined x R except at x = 9. Hence, the domain of the composite function is R - {9}.

9.
If the functions are defined by $f$($x$) = 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, ∞)

#### Solution:

f(x) = x + 5 and g(x) = | x + 3 |

(fg)(x) = f(x)g(x)
[Use the quotient rule of functions.]

= x + 5|x + 3|
[Use f(x) = x + 5 and g(x) = | x + 3 |.]

The composite function is not defined when x = - 3 and x < - 5.

Hence, the domain is [ - 5, - 3) (- 3, ∞).

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

#### Solution:

x = t6 and y = t - 35

t = y + 35
[Write t in terms of y.]

x = (y + 35)6
[Substitute t = y + 35 in x = t6.]