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Chapter 9: Problem 71

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

Short Answer

Expert verified
\( (fg)(0) = -9 \).

Step by step solution

01

Define the Functions

Given the functions: \( f(x) = x^2 - 9 \), \( g(x) = 2x \), and \( h(x) = x - 3 \).
02

Evaluate g(0)

Find the value of \( g(x) \) when \( x = 0 \). \( g(0) = 2(0) = 0 \).
03

Evaluate f(g(0))

Using the result from Step 2, evaluate \( f(g(0)) \). Since \( g(0) = 0 \), we need to find \( f(0) \).
04

Evaluate f(0)

Substitute \( x = 0 \) into \( f(x) \). \( f(0) = 0^2 - 9 = -9 \).

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Key Concepts

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

Evaluating Functions
Evaluating functions involves substituting a specific value in place of the variable in a function expression and calculating the result. For example, to evaluate the function \( f(x) = x^2 - 9 \) at \( x = 0 \), we replace \( x \) with 0 and then simplify the equation. So, \( f(0) = 0^2 - 9 = -9 \). Similarly, for the function \( g(x) = 2x \), to find \( g(0) \), we substitute 0 for \( x \) giving us \( g(0) = 2(0) = 0 \). Repeating this process allows us to evaluate any function at any given value of \( x \).
Function Notation
Function notation is a way of representing functions in mathematical form. It typically looks like \( f(x) \), where \( f \) denotes the function and \( x \) is the variable input. This notation is very useful because it clearly shows the relationship between input and output values. For instance, if we have \( f(x) = x^2 - 9 \), it tells us that for every value of \( x \), the output is the square of \( x \) minus 9. When we write \( f(3) \), it means we are evaluating the function \( f \) at \( x = 3 \). This helps to quickly understand and work with multiple functions.
Function Operations
Function operations involve combining functions in various ways鈥攕uch as addition, subtraction, multiplication, and composition. Let's look at function composition, which is used in the given exercise. Function composition means applying one function to the result of another function. Using the given functions, \( f(x) = x^2 - 9 \) and \( g(x) = 2x \), we need to find \( (f \, g)(0) \).

First, evaluate \( g(0) \):
  • \( g(x) = 2x \)
  • Substitute \( x = 0 \)
  • So, \( g(0) = 2(0) = 0 \)
Next, take this result and use it as the input for \( f(x) \):
  • \( f(x) = x^2 - 9 \)
  • Substitute \( g(0) = 0 \) into \( f \)
  • So, \( f(0) = 0^2 - 9 = -9 \)
The result of the composition \( (f \, g)(0) \) is \( -9 \). This demonstrates how function composition can be used to combine two functions by applying them sequentially.

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Most popular questions from this chapter

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (h \circ f)\left(\frac{1}{2}\right) $$

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