Problem 39

Question

$$ \text { If } f(x)=x^{2}+2 x+2, \text { find two functions } g \text { for which }(f \circ g)(x)=x^{2}-4 x+5 $$.

Step-by-Step Solution

Verified

Answer

The two functions g are $ g(x) = x - 3 $ and $ g(x) = -x + 5 $.

1Step 1: Understand the given composition

Given are two functions such that $ (f \circ g)(x) = f(g(x)) = x^{2} - 4x + 5 $. The goal is to find function(s) $ g(x) $ such that when $ g(x) $ is substituted into $ f(x) $, the result simplifies to $ x^2 - 4x + 5 $.

2Step 2: Analyze the composition

Recall the function $ f(x) = x^2 + 2x + 2 $. We need to find $ g(x) $ such that: $ f(g(x)) = (g(x))^2 + 2g(x) + 2 = x^2 - 4x + 5 $.

3Step 3: Substitution and identification

Let $ g(x) = x + c $ and substitute in the composition: $ f(g(x)) = f(x+c) = (x+c)^2 + 2(x+c) + 2 $. Simplify: $ = x^2 + 2cx + c^2 + 2x + 2c + 2 $.

4Step 4: Matching coefficients

Compare $ x^2 + 2cx + c^2 + 2x + 2c + 2 $ to $ x^2 - 4x + 5 $: Coefficient of $ x $ gives the equation $ 2c + 2 = -4 $ which simplifies to $ 2c = -6 \rightarrow c = -3 $. Verify by substituting $ c = -3 $ back: $ (x - 3)^2 + 2(x - 3) + 2 = x^2 - 6x + 9 + 2x - 6 + 2 = x^2 - 4x + 5 $. Hence, $ g(x) = x - 3 $.

5Step 5: Identify another possible solution

Consider $ g(x) = -(x+c) $ and substitute in the composition: $ f(-x+c) = (-x+c)^2 + 2(-x+c) + 2 $. Simplify and match the coefficients to find $ c = -5 $. Verify: $ (-(x - 5))^2 + 2(-(x - 5)) + 2 = x^2 - 4x + 5 $. Hence, $ g(x) = -x + 5 $.

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables and coefficients. These expressions exclusively involve operations of addition, subtraction, and multiplication, as well as non-negative integer exponents. A polynomial function of a single variable can be written as:

Problem 38

Problem 39

Other exercises in this chapter

Problem 37

Find formulas for $(f \circ g)(x)$ if $$ f(x)=\left\\{\begin{array}{ll} 0 & \text { if } x1 \end{array} \text { and } g(x)= \begin{cases}1 & \text { if } x1\e

View solution

Problem 38

$$ \text { If } f(x)=x^{2} \text {, find two functions } g \text { for which }(f \circ g)(x)=4 x^{2}-12 x+9 $$.

View solution

Problem 39

Prove that if $b>a>0$ and $c>0$, then $$ \frac{a+c}{b+c}>\frac{a}{b} $$