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Chapter 5: Problem 79

Let \(f(x)=3 x+1\) and \(g(x)=x^{2}-2 .\) Find the following. $$ (f+g)(5) $$

Short Answer

Expert verified
The value of \((f+g)(5)\) is 39.

Step by step solution

01

- Understand the given functions

We are given two functions: 1. \(f(x) = 3x + 1\) 2. \(g(x) = x^2 - 2\)
02

- Find the expression for \((f+g)(x)\)

The sum of the two functions \((f+g)(x)\) is found by adding them together: \[(f+g)(x) = f(x) + g(x)\] Substituting the given functions: \[(f+g)(x) = (3x + 1) + (x^2 - 2)\] Combine like terms: \[(f+g)(x) = x^2 + 3x - 1\]
03

- Substitute \(x = 5\) into the combined function

Now that we have \((f+g)(x) = x^2 + 3x - 1\), we substitute \(x = 5\): \[(f+g)(5) = 5^2 + 3(5) - 1\]
04

- Simplify the expression

Calculate the terms: \[5^2 = 25\] \[3(5) = 15\] Substitute back and simplify: \[(f+g)(5) = 25 + 15 - 1 = 39\]

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Key Concepts

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

Combining like terms
Combining like terms is a crucial step in simplifying expressions. Like terms are terms that contain the same variables raised to the same power. They can be added together.
In our example, after adding the functions \(f(x) = 3x + 1\) and \(g(x) = x^2 - 2\), we get
\[(f + g)(x) = (3x + 1) + (x^2 - 2)\]
Then, we combine like terms:
\[x^2 + 3x - 1\]
Here, \(3x\) and \(1\) are like terms with the same power of \(x\). By combining them, we form a simpler expression.

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