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Chapter 2: Problem 9

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

Short Answer

Expert verified
\( \frac{1}{12} \)

Step by step solution

01

- Understand the given functions

The given functions are: \( f(x) = x - 3 \) and \( g(x) = x^2 - x \).
02

- Substitute the given value into the functions

Substitute \( x = 4 \) into the functions: \( f(4) = 4 - 3 \) and \( g(4) = 4^2 - 4 \).
03

- Simplify each function for \(x = 4\)

Calculate the values: \( f(4) = 1 \) and \( g(4) = 16 - 4 = 12 \).
04

- Form the expression \((f / g)(x)\)

The expression \((f / g)(x)\) is given by \( \frac{f(x)}{g(x)} \).
05

- Simplify the expression

Plug in the calculated values: \( \frac{f(4)}{g(4)} = \frac{1}{12} \).

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Key Concepts

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

Function Evaluation
Function evaluation involves finding the value of a function at a specific input. For example, given a function like \( f(x) = x - 3 \), you can evaluate it by replacing the variable \( x \) with the given number.
In the exercise, we needed to evaluate \( f(x) = x - 3 \) and \( g(x) = x^2 - x \) at \( x = 4 \). To do this, simply substitute \( x = 4 \) into both functions:
\( f(4) = 4 - 3 = 1 \)
\( g(4) = 4^2 - 4 = 16 - 4 = 12 \).
This gives the values of the functions at \( x = 4 \). Function evaluation is fundamental in understanding how changes in the input affect the output.
Rational Expressions
A rational expression is a fraction in which both the numerator and the denominator are polynomials. In the context of our problem, the rational expression is \( \frac{f(x)}{g(x)} = \frac{x-3}{x^2-x} \).
To evaluate this at a specific value of \( x \), we substitute \( x = 4 \) into both the numerator and the denominator and then divide. From the previously calculated values:
\( f(4) = 1 \)
\( g(4) = 12 \)
This yields the rational expression evaluated at \( x = 4 \) as \( \frac{f(4)}{g(4)} = \frac{1}{12} \).
Rational expressions require careful handling because the denominator must not be zero, which would make the expression undefined.
Simplifying Expressions
Simplifying expressions means reducing them to their simplest form. This process often involves combining like terms, factoring, and canceling common factors in the numerator and denominator.
In our exercise, after evaluating the functions at \( x = 4 \), we have the expression \( \frac{1}{12} \). Since 1 and 12 have no common factors other than 1, the expression is already simplified.
The simplification process is important for making complex expressions easier to work with and understand. For rational expressions, always check if the numerator and the denominator have any common factors that can be canceled.

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