﻿ Function Notation Worksheet | Problems & Solutions Function Notation Worksheet

Function Notation Worksheet
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1.
Which of the following is true about the given functions $f$ ($x$) = $\frac{{x}^{2}+x-2}{x-1}$ and $g$ ($x$) = $x$ + 2? a. Both functions $f$($x$) = $g$($x$) + 1 b. Both functions $f$($x$), $g$($x$) are identical c. Both functions $f$($x$), $g$($x$) are not identical d. Both functions $f$($x$), $g$($x$) have the same domain

Solution:

f (x) = x2+x-2x-1 and g (x) = x + 2

Clearly, the function f (x) is continuous for all values of x except at x = 1. Hence, the domain of f(x) is R - {1}.

For all real x except 1, f (x) = x + 2
[Use polynomial division.]

The function g (x) is defined for all real values of x. Hence, the domain of g(x) is R.

So, f (x) = x + 2 with domain R - {1}, and g(x) = x + 2 with domain R.

Though both the functions have the same definitions , they are not identical as they have different domains.

2.
Choose the domain of the function $f$ ($x$) = . a. All real $x$ except - 16 b. All real $x$ except 16 c. All real vaues d. All real $x$ except - 17

Solution:

The domain of the function f (x) = 1x + 16 is all the real values of x except - 16
[f (x) = 1x + 16 is defined for all the real values of x except -16.]

3.
Which of the following transformations have to be applied to the graph of reciprocal function $f$ ($x$) = $\frac{1}{x}$ to get the graph of $g$ ($x$) = ? a. Translation to the left by 2 units and a vertical stretch by a factor of 6. b. Translation to the left by 6 units and a vertical stretch by a factor of 6. c. Translation to the left by 6 units and a vertical stretch by a factor of 2. d. Translation to the left by 2 units and a vertical stretch by a factor of 2.

Solution:

f (x) = 1x and g (x) = 2x + 6

g (x) = 2x + 6

= 2 (1x + 6)

= 2 · f (x + 6) where f (x) = 1x

The graph of g is the graph of the reciprocal function f translated to the left by 6 units then stretched vertically by a factor of 2.

4.
Which of the following transformations have to be applied to the graph of reciprocal function $f$ ($x$) = $\frac{1}{x}$ to get the graph of $g$ ($x$) = ? a. Translation to the right by 5 unit and then translation vertically up by 6 units. b. Translation to the right by 6 unit and then translation vertically up by 5 units. c. Translation to the right by 6 unit and then translation vertically up by 6 units. d. Translation to the right by 5 unit and then translation vertically up by 5 units.

Solution:

f (x) = 1x and g (x) = 5x - 29x - 6

g (x) = 5x - 29x - 6

g (x) = 5 + 1x - 6
[Use polynomial division.]

So, g (x) = 5 + f (x - 6) Where f (x) = 1x.

The graph of g is the graph of the reciprocal function f translated to the right by 6 unit and then translated up by 5 units.

5.
Which of the following is the vertical asymptote of $h$($x$) = $\frac{{x}^{2}+36}{x}$? a. $y$ = 0 b. $x$ = - 36 c. $x$ = 36 d. $x$ = 0

Solution:

f (x) = x2+36x

x = 0
[Find the zeros of the denominator.]

The vertical asymptote is the line x = 0.
[For the graph of a rational function vertical asymptotes occur at the zeros of the denominator, which are not the zeros of the numerator.]

6.
Find the $x$ - intercept of the graph of $f$ ($x$) = $\frac{{x}^{2}-9x+20}{x+5}$. a. (4 , 0) and (- 5, 0) b. (- 4 , 0) and (- 5, 0) c. (- 4, 0) and (5, 0) d. (4 , 0) and (5, 0)

Solution:

f (x) = x2-9x+20x+5

x2 - 9x + 20 = 0
[Solve the numerator for the zeros.]

(x - 5)(x - 4) = 0
[Factor.]

x - 5 = 0 or x - 4 = 0

The zeros of the numerator are x = 4 and 5

So, the x - intercepts of the graph of the given function are (4, 0) and (5, 0).
[The x - intercepts of the graph of a rational function are the zeros of its numerator.]

7.
Which of the following would be the end behavior asymptote of $y$ = ? a. $y$ = $x$ + 25 b. $y$ = $x$ - 5 c. $y$ = $x$ + 5 d. $y$ = 5$x$

Solution:

y = x2+25x - 5

y = (x + 5) + 50x - 5
[Use polynomial division.]

Hence, the end behaviour asymptote of the graph of the given function is y = x + 5
[The end behaviour asymptote of y = q (x) + r(x)g(x) is y = q (x).]

8.
Which of the following transformations have to be applied to the graph of reciprocal function $f$ ($x$) = $\frac{1}{x}$ to obtain the graph of $g$ ($x$) = ? a. Translation to the right by 10 units followed by a reflection about $y$-axis. b. Translation to the right 10 units, vertical stretch by a factor of 10, a reflection across $x$ - axis and translation vertically down by 9 units. c. Translation to the right 10 units, vertical stretch by a factor of 2 , a reflection across $x$ - axis and translation vertically down by 9 units. d. Translation to the right 10 units, vertical stretch by a factor of 2 , a reflection across $x$ - axis and translation vertically down by 2 units.

Solution:

f (x) = 1x and g (x) = 88 - 9xx - 10

g (x) = 88 - 9xx - 10

g(x) = - 9 - 2x - 10
[Use polynomial division.]

= - 9 - 2(1x - 10)

= - 9 - 2 f (x - 10) where f (x) = 1x

The graph of g(x) can be obtained by a translation to the right 10 units, vertical stretch by a factor of 2, a reflection across the x - axis and then a translation vertically down by 9 units on the graph of f(x).

9.
Use the graph of the function $f$ ($x$) to evaluate $\underset{x\to \infty }{\mathrm{lim}}$ $f$ ($x$).  a. 3 b. ∞ c. -∞

Solution:

As the value of x is increasing without any bound, the value of f (x) is approaching the zero.
[From the graph.]

Hence, limx f (x) = 0.

10.
Use the graph of the function $f$ ($x$) to evaluate .  a. -∞ b. ∞ c. 3

Solution:

Evidently, from the graph, the curve is discontinuous at x = -2.

As x -2-, the value of f (x) is approaching a very large value (∞).
[From the graph.]

Hence, limx-2- f(x) = ∞.