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Chapter 4: Problem 40

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. \((g-h)(10)\)

Short Answer

Expert verified
(g-h)(10) = 13

Step by step solution

01

Understand the Problem

The problem asks for the value of (g-h)(10), where g(x)=2x and h(x)=x-3.
02

Define the Combined Function

Recall that (g-h)(x) is defined as g(x)-h(x). First, find the expressions for g(x) and h(x):g(x)=2x h(x)=x-3. Then, subtract them:(g-h)(x) = g(x) - h(x) = 2x - (x-3).
03

Simplify the Combined Function

Simplify the combined function:(g-h)(x) = 2x - (x-3) = 2x - x +3 = x + 3.
04

Evaluate at Specific Value

Plug in the value x = 10 into the simplified function:(g-h)(10) = 10 + 3 = 13.

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Key Concepts

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

Evaluating Functions
In algebra, evaluating functions involves finding the output of a function for a specific input value. For example, if you have a function defined as \(f(x) = x^2 - 9\), and you want to evaluate it at \(x = 2\), you would replace every instance of \(x\) in the function with 2. So, \(f(2) = 2^2 - 9 = 4 - 9 = -5\).
Here's a step-by-step process to evaluate any function:
  • Identify the function's definition (e.g., \(f(x) = x^2 - 9\)).
  • Substitute the given value for every instance of \(x\) in the function (e.g., for \(x = 2\)).
  • Simplify the resulting expression to find the output (e.g., \(2^2 - 9 = -5\)).
By following these steps, you can evaluate any function at any input value. In our original problem, we evaluated \((g-h)(10)\) after simplifying the combined function.
Function Subtraction
Function operations include addition, subtraction, multiplication, and division of two or more functions. When you subtract one function from another, you write the new function definition that subtracts the expressions. Let's break down the steps for function subtraction:
  • Start with the definitions of the two functions you will subtract. In our example, \(g(x) = 2x\) and \(h(x) = x - 3\).
  • Set up the subtraction: \((g-h)(x) = g(x) - h(x)\).
  • Replace \(g(x)\) and \(h(x)\) with their definitions: \((g-h)(x) = 2x - (x - 3)\).
  • Simplify the resulting expression: \(2x - x + 3\).
The result is the new function, \((g-h)(x) = x + 3\). You then use this new function to find the value for the given input鈥攊n our case, \(x=10\).
Simplifying Expressions
Simplifying expressions is an essential skill in algebra. It means rewriting an expression in its simplest form. This often involves combining like terms, factoring, and performing arithmetic operations. Here's how you can simplify expressions step-by-step:
  • Identify and group like terms. For example, in the expression \(2x - (x - 3)\), both \(2x\) and \(x\) are like terms.
  • Apply the distributive property if necessary. In our example, \(-1(x - 3)\) becomes \(-x + 3\).
  • Combine the like terms: \(2x - x + 3\)
  • Simplify to get the simplest form: \(x + 3\).
In our problem, after simplifying the expression \(2x - (x - 3)\), we got \(x + 3\). We then evaluated this new function at \(x = 10\), resulting in 13. Simplifying expressions helps in reducing the complexity of problems, making them easier to solve.

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Most popular questions from this chapter

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