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Chapter 5: Problem 77

Let \(f(x)=3 x+1\) and \(g(x)=x^{2}-2 .\) Find the following. $$ g(-10) $$

Short Answer

Expert verified
g(-10) = 98

Step by step solution

01

Identify the function definition

The function given is \(g(x) = x^2 - 2\).
02

Substitute the value into the function

Substitute \(x = -10\) into the function \(g(x) = x^2 - 2\).
03

Simplify the expression

Calculate \(g(-10) = (-10)^2 - 2\). This equals \(100 - 2\).
04

Compute the final value

Subtract \(2\) from \(100\) to get \(98\). Therefore, \(g(-10) = 98\).

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Key Concepts

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

Function Definition
To solve problems related to functions, you first need to understand what a function is. A function is like a machine that takes an input and produces an output according to a specific rule. In mathematical terms, we write a function as \(f(x)\), where \(f\) is the name of the function, and \(x\) is the variable or input. For example, the function \(g(x) = x^2 - 2\) takes any number \(x\), squares it, and then subtracts 2. This rule helps us determine the output for any given input.
Substitution
Substitution involves replacing the variable in a function with a specific value. In our problem, we need to find \(g(-10)\). This means we substitute \(-10\) in place of \(x\) in the function \(g(x) = x^2 - 2\). When substituting, carefully follow the function's rule using the new value. So, we replace \(x\) with \(-10\) and get \(g(-10) = (-10)^2 - 2\).
Simplification
Simplification is the process of performing mathematical operations to make an expression easier to understand. After substituting \(-10\) into our function \(g(x)\), we get the expression \(g(-10) = (-10)^2 - 2\). The next step is to simplify it by computing the square of \(-10\). Remember that squaring a negative number always gives a positive result: \ \((-10)^2 = 100\). So, the expression simplifies to \(100 - 2\).
Algebraic Operations
To complete the problem, we need to perform the final algebraic operation, which is subtraction. From the simplified expression \(100 - 2\), we subtract \(2\) from \(100\). This step gives us the result \(98\). Therefore, \(g(-10) = 98\).
  • Substituted value: \(-10\)
  • Simplified expression: \(100 - 2\)
  • Final result: \(98\)
This way, we use algebraic operations to arrive at the final value.

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