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Chapter 4: Problem 113

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

Short Answer

Expert verified
(fg)(2) = -20

Step by step solution

01

Definition of function products

The product of two functions, denoted as \( fg \), is defined as \( (fg)(x) = f(x) \cdot g(x) \).
02

Evaluate each function at x = 2

First, find the value of \ f(2) \ and \ g(2) \ individually. \[ f(2) = 2^2 - 9 = 4 - 9 = -5 \] \[ g(2) = 2 \cdot 2 = 4 \]
03

Multiply the results

Now multiply the values obtained for \ f(2) \ and \ g(2) \: \[ (fg)(2) = f(2) \cdot g(2) = -5 \cdot 4 = -20 \]

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Key Concepts

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

definition of function products
Understanding how to multiply functions is crucial in algebra. When we talk about the product of two functions, we mean combining them in a specific way.
This process is denoted as \((fg)(x)\).
  • To multiply two functions, say \(f\) and \(g\), you do it point-wise.
  • In other words, you find the value of both functions at a given \(x\), then multiply those values together.
\( (fg)(x) = f(x) \cdot\ g(x) \) is the formal way we show this relationship.
This means that for any input \(x\), the output of the product function is the product of the outputs of \(f(x)\) and \(g(x)\). Understanding this definition paves the way for more advanced concepts in algebra and calculus.
evaluating functions
Now that you understand how to define function products, let's move forward to evaluating functions. Evaluating a function simply means finding the output value for a specific input value.
Let's break it down:
  • Given a function \(f(x)\), finding \(f(a)\) means placing the value \('a'\) into \('x'\) in the function's formula.
For example, if \ f(x) = x^2 - 9 \, then to evaluate \ f(2) \ you replace \ x \ with \ 2 \: \ f(2) = 2^2 - 9 = 4 - 9 = -5 \.
It's like following a recipe where you swap variables for actual numbers to get the result. Evaluating functions is a stepping stone to more complex algebraic manipulations.
multiplying functions at given points
For the final step, we use the concepts we have learned to multiply functions at a specific point. This involves two main tasks: first evaluating each function at given points and then multiplying the resulting values.
  • Step 1: Evaluate both functions at \(x=2\).
  • For \(f(x) = x^2-9\), calculate \(f(2)\).
  • For \(g(x) = 2x\), calculate \( g(2) \).
So, \ f(2) = 2^2 - 9 = -5 \ and \ g(2) = 2 \cdot\ 2 = 4 \.
Now that we have our values, let's multiply them:
\ (fg)(2) = f(2) \cdot\ g(2) = -5 \cdot\ 4 = -20 \.
By following these steps, you can multiply any two functions at any given point.

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