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Chapter 9: Problem 72

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following $$ (f h)(0) $$

Short Answer

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Step by step solution

01

Find h(0)

First, substitute 0 into the function h(x). So, we have: h(0) = 0 - 3 h(0) = -3
02

Substitute h(0) into f(x)

Next, take the result from Step 1, which is -3, and substitute it into the function f(x). f(h(0)) = f(-3) f(x) = x^{2} - 9 f(-3) = (-3)^{2} - 9 f(-3) = 9 - 9 f(-3) = 0

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Key Concepts

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

Understanding Function Notation
Function notation is a way to express the output of a function in terms of the input. For example, in the function \( f(x) = x^2 - 9 \), \( f(x) \) represents the output when \( x \) is the input. Function notation allows us to easily substitute different values for \( x \) and find corresponding values of \( f(x) \).

Here鈥檚 how it works:
  • The letter \( f \) represents the function name, while \( x \) is the variable or input.
  • The expression after the equal sign shows the calculation to be performed on \( x \).
By using function notation, we get a clear and concise way to define and evaluate functions. It helps keep things organized, especially when dealing with more complex operations like composite functions.
The Substitution Process
Substitution is a fundamental technique used in math to replace variables with actual numbers. Let's see how it applies to our problem: We first need to evaluate \( h(0) \). Given \( h(x) = x - 3 \), we substitute 0 for \( x \):

$$ h(0) = 0 - 3 $$ $$ h(0) = -3 $$
Once we have the value of \( h(0) \), which is -3, we use this result in the next function, \( f(x) \). We substitute \( -3 \) for \( x \) in \( f(x) = x^2 - 9 \):

$$ f(-3) = (-3)^2 - 9 $$ $$ f(-3) = 9 - 9 $$ $$ f(-3) = 0 $$
Substitution helps break down complex problems into manageable steps. By solving each part sequentially, we avoid errors and ensure accuracy.
Performing Algebraic Operations
Algebraic operations involve calculations like addition, subtraction, multiplication, and exponentiation. In our solution, we applied these operations as follows:
  • First, in the function \( h(x) = x - 3 \), we used subtraction to find \( h(0) = -3 \).
  • Next, we substituted \( -3 \) into \( f(x) = x^2 - 9 \) and performed exponentiation (raising to the power of 2) and subtraction.
Here are more details on these operations:

Exponentiation: When we calculated \( (-3)^2 \), we squared -3, giving us 9.
Subtraction: In both \( h(0) = 0 - 3 \) and \( f(-3) = 9 - 9 \), we subtracted one number from another.

These algebraic steps are essential building blocks in evaluating functions, especially in composite functions where one function's output becomes another's input.

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Most popular questions from this chapter

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