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Chapter 2: Q. 3 (page 124)

Find the following for each function:
$f (x) = \frac{3 x}{x^{2} - 1}$
Part (a): $f (2)$
Part (b): $f (- 2)$
Part (c): $f (- x)$
Part (d): $- f (x)$
Part (e): $f (x - 2)$
Part (f): $f (2 x)$

Short Answer

Expert verified

Part (a): $f (2) = 2$

Part (b): $f (- 2) = - 2$

Part (c): $f (- x) = \frac{- 3 x}{x^{2} - 1}$

Part (d): $- f (x) = \frac{- 3 x}{- x^{2} + 1}$

Part (e): $f (x - 2) = \frac{3 (x - 2)}{x^{2} - 4 x + 3}$

Part (f): $f (2 x) = \frac{6 x}{4 x^{2} - 1}$

Step by step solution

01

Part (a) Step 1. Given information.

Consider the given function,

$f (x) = \frac{3 x}{x^{2} - 1}$

Substitute $x = 2$ in the given equation,

localid="1645858263721" $f (2) = \frac{3 (2)}{{(2)}^{2} - 1} f (2) = \frac{6}{4 - 1} f (2) = \frac{6}{3} f (2) = 2$

02

Part (b) Step 1. Given information.

Consider the given function,

$f (x) = \frac{3 x}{x^{2} - 1}$

Substitute $x = - 2$ in the given equation,

$f (- 2) = \frac{3 (- 2)}{{(- 2)}^{2} - 1} f (- 2) = \frac{- 6}{4 - 1} f (- 2) = \frac{- 6}{3} f (- 2) = - 2$

03

Part (c) Step 1. Given information.

Consider the given function,

$f (x) = \frac{3 x}{x^{2} - 1}$

Substitute $x = - x$ in the given equation,

$f (- x) = \frac{3 (- x)}{{(- x)}^{2} - 1} f (- x) = \frac{- 3 x}{x^{2} - 1}$

04

Part (d) Step 1. Given information.

Consider the given function,

$f (x) = \frac{3 x}{x^{2} - 1}$

From the question,

$- f (x) = - (\frac{3 x}{x^{2} - 1}) - f (x) = \frac{- 3 x}{- x^{2} + 1}$

05

Part (e) Step 1. Given information.

Consider the given function,

$f (x) = \frac{3 x}{x^{2} - 1}$

Substitute $x - 2$ in the given equation,

$f (x - 2) = \frac{3 (x - 2)}{{(x - 2)}^{2} - 1} f (x - 2) = \frac{3 (x - 2)}{x^{2} + 4 - 4 x - 1} f (x - 2) = \frac{3 (x - 2)}{x^{2} - 4 x + 3}$

06

Part (f) Step 1. Given information.

Consider the given function,

$f (x) = \frac{3 x}{x^{2} - 1}$

Substitute $2 x$ in the given equation,

$f (2 x) = \frac{3 (2 x)}{{(2 x)}^{2} - 1} f (2 x) = \frac{6 x}{4 x^{2} - 1}$

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Most popular questions from this chapter

In Problems 7–18, match each graph to one of the following functions:

$A . y = x^{2} + 2 B . y = - x^{2} + 2 C . y = |x| + 2 D . y = - |x| + 2 E . y = {(x - 2)}^{2} F . y = - {(x + 2)}^{2} G . y = |x - 2| H . y = - |x + 2| I . y = 2 x^{2} J . y = - 2 x^{2} K . y = 2 |x| L . y = - 2 |x|$

A right triangle has one vertex on the graph of $y = x^{3}, x > 0,$ at $(x, y)$ , another at the origin, and the third on the positive y-axis at $(0, y)$ , as shown in the figure. Express the area Aof the triangle as a function of x.

If $f (x) = \{\begin{cases} - 3 x i f x < - 1 \\ 0 i f x = - 1 \\ 2 x^{2} + 1 i f x > - 1 \end{cases}$

Find: (a) $f (- 2)$ and (b) $f (- 1)$ (c) $f (0)$

Determine whether each relation represents a function. For each function, state the domain and range.

$\{(- 1, 0), (2, 3), (4, 0)\}$

Suppose that the graph of a function $g$ is known. The graph of $y = g (x) + 2$ may be obtained by a _______ shift of the graph of $g$ ________ a distance of $2$ units.

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