In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(x) = |x+3|/|x + 3| a. f(5)

Verified step by step guidance

1. Step 1: Understand the function f(x) = |x + 3| / |x + 3|. This function involves absolute values in both the numerator and denominator. The absolute value of a number is its distance from zero, so it is always non-negative.
2. Step 2: Substitute the given value of x = 5 into the function. This means replacing x with 5 in both the numerator and denominator: f(5) = |5 + 3| / |5 + 3|.
3. Step 3: Simplify the expression inside the absolute value symbols. Calculate 5 + 3, which equals 8. So the function becomes f(5) = |8| / |8|.
4. Step 4: Evaluate the absolute values. Since 8 is already positive, |8| = 8. Therefore, the function simplifies to f(5) = 8 / 8.
5. Step 5: Simplify the fraction. Divide the numerator by the denominator: 8 / 8 simplifies to 1. Thus, the value of f(5) is 1.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, represents the distance of a number x from zero on the number line, always yielding a non-negative result. For example, |5| equals 5, while |-5| also equals 5. This function is crucial in evaluating expressions that involve both positive and negative values.

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable in a function to determine the corresponding output. In this case, evaluating f(5) means replacing x with 5 in the function f(x) = |x + 3| / |x + 3|, which simplifies the process of finding the function's value at that point.

Simplification of Rational Expressions

Simplification of rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. In the given function, |x + 3| appears in both the numerator and denominator, which can lead to simplification, provided that the denominator is not zero, ensuring the expression remains valid.