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Understanding absolute value is essential when evaluating certain types of functions. The absolute value of a number is its distance from zero on the number line, regardless of the direction. It is always a positive number or zero. Absolute value is represented by two vertical bars, for example, \( |x+3| \). This means 'regardless of whether \( x+3 \) is positive or negative, take its positive value.'

In the given function \( f(x) = \frac{|x+3|}{x+3} \), the absolute value affects the numerator. Depending on the value of \( x \), the expression inside the absolute value can be negative or positive. The purpose of the absolute value is to ensure that the numerator is always non-negative.

• If \( x+3 > 0 \), then \( |x+3| = x+3 \).
• If \( x+3 < 0 \), then \( |x+3| = -(x+3) \).

Piecewise functions are defined by multiple sub-functions, each applying to different parts of the function's domain. For example, \( f(x) = \frac{|x+3|}{x+3} \) can act like a piecewise function based on the value of \( x \). This means the function can take on different expressions depending on the input value.

This specific function has two cases because of the absolute value in the numerator:

• When \( x+3 > 0 \), the function simplifies to \( f(x) = 1 \), as the numerator and the denominator are equal and positive.
• When \( x+3 < 0 \), the function simplifies to \( f(x) = -1 \), because the numerator \( |x+3| \) is positive and the denominator \( x+3 \) is negative, making the overall fraction negative.

Understanding piecewise functions helps you anticipate and verify the behavior of complex functions.

Evaluating a function means substituting specific values for the independent variable, commonly \( x \), to find the corresponding function's output. In the exercise, you are given specific values to substitute into the function \( f(x) = \frac{|x+3|}{x+3} \).

Here's how you evaluate:

• Identify the given value for the independent variable \( x \).
• Substitute this value into the function, replacing all instances of \( x \) with the given number.
• Perform arithmetic operations as needed, following the order of operations: absolute values, multiplication and division, etc.

For example:

• To find \( f(5) \), replace \( x \) with \( 5 \), resulting in \( f(5) = \frac{|5+3|}{5+3} = \frac{8}{8} = 1 \).
• For \( f(-5) \), replace \( x \) with \( -5 \), giving \( f(-5) = \frac{|-5+3|}{-5+3} = \frac{2}{-2} = -1 \).
• The result may be affected by the structure of the function, like absolute values or potential piecewise definitions, as seen in \( f(-9-x) \).

Correctly following these steps ensures that you accurately compute the function for any given value of \( x \).