Problem 5 Let $f(x)=x^{2}+3$ and \(g(x)=... [FREE SOLUTION]

91影视

Precalculus Student Solutions Manual 5th

Margaret L. Lial, John Hornsby, David I. Schneider

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 2: Problem 5

Let $f(x)=x^{2}+3$ and $g(x)=-2 x+6 .$ Find each of the following. $$(f g)(4)$$

Short Answer

Expert verified

(fg)(4) = 7

Step by step solution

- Evaluate g(4)

First, find the value of the function g at x = 4. Given: \[ g(x) = -2x + 6 \] Plug in x = 4: \[ g(4) = -2(4) + 6 \] Simplify: \[ g(4) = -8 + 6 \] \[ g(4) = -2 \]

- Evaluate f(-2)

Next, use the value of g(4) found in Step 1 to find the value of the function f at this result. Given: \[ f(x) = x^{2} + 3 \] Since g(4) = -2, plug in x = -2: \[ f(-2) = (-2)^{2} + 3 \] Simplify: \[ f(-2) = 4 + 3 \] \[ f(-2) = 7 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

f(x)=x^2+3

In this section, we look at the function $ f(x) = x^2 + 3 $.

Functions describe how inputs are transformed into outputs. Here, $ f(x) $ takes any number $ x $ and squares it first. Then, it adds 3 to the squared result.

Let's break it down step by step:

Take the input $ x $.
Square $ x $ to obtain $ x^2 $.
Add 3 to $ x^2 $ to get the final result.

Example: Let's evaluate $ f(2) $.

First, we calculate $ 2^2 $, which equals 4.
Next, we add 3 to 4, which gives us 7.

So, $ f(2) = 7 $.

This function shows us a quadratic relationship, meaning the outputs grow faster as our inputs get larger.

g(x)=-2x+6

The function $ g(x) = -2x + 6 $ has a different structure. Here鈥檚 how we can break it down:

First, take the input $ x $.
Multiply it by -2.
Add 6 to the product.

This is a linear function, which means its graph is a straight line. Each step multiplies $ x $ by a negative number (-2), causing the line to slope downward as $ x $ increases.

Example: Let's evaluate $ g(3) $:

First, multiply 3 by -2. This results in -6.
Next, add 6 to -6. This gives us 0.

So, $ g(3) = 0 $.

Linear functions increase or decrease at a constant rate.

evaluate functions

To evaluate functions, we substitute the input value into the function and perform the operations. Let's use the original exercise:

1. Evaluate $ g(4) $:
- Given $ g(x) = -2x + 6 $, substitute $ x = 4 $.

$ g(4) = -2(4) + 6 = -8 + 6 = -2 $

So, $ g(4) = -2 $.

2. Evaluate $ f(-2) $ using the result of $ g(4) $:
- Given $ f(x) = x^2 + 3 $, substitute $ x = -2 $.

$ f(-2) = (-2)^2 + 3 = 4 + 3 = 7 $

So, $ f(-2) = 7 $.

This illustrates how you can combine functions and use the result of one as the input for another.

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91影视

Let \(f(x)=x^{2}+3\) and \(g(x)=-2 x+6 .\) Find each of the following. $$(f g)(4)$$

Short Answer

Step by step solution

- Evaluate g(4)

- Evaluate f(-2)

Key Concepts

f(x)=x^2+3

g(x)=-2x+6

evaluate functions

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