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Chapter 2: Problem 10

Let \(f(x)=x-3\) and \(g(x)=x^{2}-x .\) Find and simplify each expression. $$(g / f)(4)$$

Short Answer

Expert verified
The value of \((g / f)(4)\) is \(12\).

Step by step solution

01

Understand the problem

Given two functions, \(f(x) = x - 3\) and \(g(x) = x^2 - x\). The problem asks you to find and simplify the expression \((g / f)(4)\). This represents the value of the function \(g(x)\) divided by the function \(f(x)\) evaluated at \(x = 4\).
02

Evaluate \(f(x)\) at \(x = 4\)

Calculate \(f(4)\) using the function \(f(x) = x - 3\). Substituting \(x = 4\), we get: \[ f(4) = 4 - 3 = 1 \].
03

Evaluate \(g(x)\) at \(x = 4\)

Calculate \(g(4)\) using the function \(g(x) = x^2 - x\). Substituting \(x = 4\), we get: \[ g(4) = 4^2 - 4 = 16 - 4 = 12 \].
04

Divide \(g(4)\) by \(f(4)\)

Now that we have \(g(4) = 12\) and \(f(4) = 1\), we divide \(g(4)\) by \(f(4)\): \[ (g / f)(4) = \frac{g(4)}{f(4)} = \frac{12}{1} = 12 \].

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Key Concepts

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

Function Evaluation
When we talk about function evaluation, we mean finding the value of a function for a specific input.
A function is like a machine that produces an output based on the input you give it.
Here's how you can evaluate a function:
  • Identify the input value.
  • Replace the variable in the function with this input value.
  • Simplify the expression to find the output.
This method helps us understand what happens to the input as it travels through the function. In the previous example, we had two functions:
f(x) = x - 3 and g(x) = x^2 - x.
We evaluated these functions at a specific point, x = 4. This allowed us to find the values
f(4) and g(4), which were 1 and 12 respectively.
Function Division
Function division involves dividing the output of one function by the output of another function.
This is also called the quotient of the functions. To divide functions, follow these steps:
  • Find the value of each function at the given input (function evaluation).
  • Divide the output of the first function by the output of the second function.
  • Simplify the result if possible.
This concept is useful when you want to understand the relationship between two functions.
In our example, we found g(4) and f(4) and then calculated the division as:
\[\frac{g(4)}{f(4)} = \frac{12}{1} = 12 \]
Composite Functions
Composite functions involve applying one function to the result of another function.
Different from the previous concepts, this one chains two functions together.
If we have two functions, say f(x) and g(x), the composite function (g 鈭� f)(x) is defined as g(f(x)).
Here's how you find a composite function:
  • Start with an input value.
  • First, apply the inner function (here, f(x)) to the input value.
  • Then, take the result and apply the outer function (here, g(x)) to this result.
  • Simplify your final answer.
Composite functions illustrate how outputs from one function can feed into another, creating a pipeline effect.
If we had been asked to compute (g 鈭� f)(x) at some point, we would first find f(x) and then use this result as the input for g(x).

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Most popular questions from this chapter

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=|x-2|$$

Solve each inequality by graphing an appropriate function. State the solution set using interval notation. $$5-\sqrt{x} \geq 0$$

Make a table listing ordered pairs for each function. Then sketch the graph and state the domain and range. Identify any intervals on which \(f\) is increasing, decreasing, or constant. $$f(x)=|x|+1$$

Use transformations to graph each function and state the domain and range. $$y=-4 x+200$$

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=x^{4}-x^{3}$$
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