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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 7: Q. 12 (page 506)

If $f (x) = 2 x^{2} + 3 x + 1$ and $g (x) = x^{2} + 3 x + 2$ , solve:
Part (a): $f (x) = 0$
Part (b): $f (x) = g (x)$
Part (c): $f (x) > 0$
Part (d): $f (x) 鈮� g (x)$

Short Answer

Expert verified

Part (a): When $f (x) = 0$ , then $x = \{- \frac{1}{2}, - 1\}$ .

Part (b): When $f (x) = g (x)$ , then $x = \{- 1, 1\}$ .

Part (c): When $f (x) > 0$ , then role="math" localid="1646574659662" $x = (鈭�, - 1) 鈭� (- \frac{1}{2}, 鈭�)$ .

Part (d): When $f (x) 鈮� g (x)$ , then $x = (- 鈭�, - 1] 鈭� [1, 鈭�)$ .

Step by step solution

01

Part (a) Step 1. Solve the equation when fx=0.

Consider the given question,

$f (x) = 2 x^{2} + 3 x + 1, g (x) = x^{2} + 3 x + 2$

From the question,

$f (x) = 0 2 x^{2} + 3 x + 1 = 0 2 x^{2} + 2 x + x + 1 = 0 2 x (x + 1) + (x + 1) = 0 (2 x + 1) (x + 1) = 0 x = \{- \frac{1}{2}, - 1\}$

02

Part (b) Step 1. Solve the equation when fx=gx.

Consider the given question,

$f (x) = 2 x^{2} + 3 x + 1, g (x) = x^{2} + 3 x + 2$

From the question,

$f (x) = g (x) 2 x^{2} + 3 x + 1 = x^{2} + 3 x + 2 x^{2} - 1 = 0 (x + 1) (x - 1) = 0 x = 卤 1 x = \{- 1, 1\}$

03

Part (c) Step 1. Solve the equation when fx>0.

Consider the given question,

$f (x) = 2 x^{2} + 3 x + 1, g (x) = x^{2} + 3 x + 2$

From the question,

$f (x) > 0 2 x^{2} + 3 x + 1 > 0 (2 x + 1) (x + 1) > 0 x = (鈭�, - 1) 鈭� (- \frac{1}{2}, 鈭�)$

04

Part (d) Step 1. Solve the equation when fx≥gx.

Consider the given question,

$f (x) = 2 x^{2} + 3 x + 1, g (x) = x^{2} + 3 x + 2$

From the question,

$f (x) 鈮� g (x) 2 x^{2} + 3 x + 1 鈮� x^{2} + 3 x + 2 2 x^{2} - x^{2} + 3 x - 3 x + 1 - 2 鈮� 0 x^{2} - 1 鈮� 0 x = (- 鈭�, - 1] 鈭� [1, 鈭�)$

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