In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(x) = |x+3|/|x + 3| b. 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 expressions inside the absolute value bars. For the numerator and denominator, calculate (-5) + 3, which equals -2. So the function becomes f(-5) = |-2| / |-2|.
4. Step 4: Evaluate the absolute values. The absolute value of -2 is 2, so the function simplifies to f(-5) = 2 / 2.
5. Step 5: Simplify the fraction. Divide the numerator by the denominator to find the simplified value of the function at x = -5.

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, |3| = 3 and |-3| = 3. This function is crucial in evaluating expressions that involve both positive and negative values, as it affects the outcome based on the sign of the input.

Evaluating Functions

Evaluating a function involves substituting a specific value for the independent variable (in this case, x) into the function's expression. This process allows us to determine the corresponding output value. For instance, to evaluate f(-5), we replace x with -5 in the function f(x) = |x + 3| / |x + 3| and simplify the expression accordingly.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form, often by performing arithmetic operations and applying algebraic rules. In the context of the given function, simplification may include canceling common factors or reducing fractions. Understanding how to simplify is essential for accurately determining the value of the function after evaluation.