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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 4: Q. 14 (page 247)

For the function $f (x) = x^{2} + 5 x - 2$
Find
$a) f (3) b) f (- x) c) - f (x) d) f (3 x) e) \frac{f (x + h) - f (x)}{h}$

Short Answer

Expert verified

The solution is

$a) f (3) = 22 b) f (- x) = x^{2} - 5 x - 2 c) - f (x) = - x^{2} - 5 x + 2 d) f (3 x) = 9 x^{2} + 15 x - 2 e) \frac{f (x + h) - f (x)}{h} = 2 x + h + 5$

Step by step solution

01

Given information

We are given a equation $f (x) = x^{2} + 5 x - 2$

02

Part a) Step 1: Substitute x=3 in the given equation

We get,

$f (x) = x^{2} + 5 x - 2 f (3) = 9 + 15 - 2 f (3) = 22$

03

part b) Step 1: To find f(-x) we substitute -x in the function

We get

$f (x) = x^{2} + 5 x - 2 f (- x) = x^{2} - 5 x - 2$

04

Part c) Step 1: Find -f(x)

We get,

$f (x) = x^{2} + 5 x - 2 - f (x) = - x^{2} - 5 x + 2$

f (x) = x^{2} + 5 x - 2 - f (x) = - x^{2} - 5 x + 2

05

part d) Step 1: Find f(3x)

We get,

$f (x) = x^{2} + 5 x - 2 f (3 x) = 9 x^{2} + 15 x - 2$

06

Part e) Step 1: Find f(x+h)-f(x)h

We get $\frac{f (x + h) - f (x)}{h} \frac{{(x + h)}^{2} + 5 (x + h) - 2 - x^{2} - 5 x + 2}{h} \frac{x^{2} + 2 x h + h^{2} + 5 x + 5 h - 2 - x^{2} - 5 x + 2}{h} \frac{2 x h + h^{2} + 5 h}{h} 2 x + h + 5$ $\frac{f (x + h) - f (x)}{h} \frac{{(x + h)}^{2} + 5 (x + h) - 2 - x^{2} - 5 x + 2}{h} \frac{x^{2} + 2 x h + h^{2} + 5 x + 5 h - 2 - x^{2} - 5 x + 2}{h} \frac{2 x h + h^{2} + 5 h}{h} 2 x + h + 5$

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