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Breaking down a function evaluation into manageable steps can make the process simpler. Let's use the function \( h(z) = \frac{1}{z+4} \) as an example to find \( h(-5) \). The first thing to understand is the procedure for plugging in the value provided in the exercise.

Start by replacing the variable in the function with the given number. Here, replace the \( z \) with \( -5 \). This substitution leads to the expression \( h(-5) = \frac{1}{-5+4} \). This is our first step, called 'Substitution'.

The next step is to solve the denominator, which involves basic arithmetic. Compute the value: \( -5 + 4 \). Performing this simple subtraction gives us \( -1 \). Substituting this back into the original expression simplifies the function to \( h(-5) = \frac{1}{-1} \).

Finally, simplify the resulting fraction. Here, the fraction \( \frac{1}{-1} \) simplifies to \(-1\). Thus, this step-by-step approach allows us to conclude that \( h(-5) = -1 \).

Following these steps provides clarity and understanding of how to evaluate functions effectively.

The substitution method is crucial in function evaluation. It refers to replacing a variable with a specific number to make calculations easier. For instance, in finding \( h(-5) \) for the function \( h(z) = \frac{1}{z+4} \), you substitute \( z \) with \(-5\).

This method consists of a few simple steps:

• Identify the variable that needs substitution. Here, it's \( z \).
• Replace as specified in the problem—\( z = -5 \).
• Rewrite the function using this substitution: \( h(-5) = \frac{1}{-5+4} \).

This makes the process systematic and provides a clear pathway to proceed with solving or simplifying any equations that result from substitution.

Using substitution is not limited to single-variable functions only. It can also be applied to equations with several variables in more complex problems. It's an essential tool in mathematics, allowing for easier computation and problem-solving.

Fraction simplification is an important skill in both basic and advanced mathematics. In the context of functions like \( h(z) = \frac{1}{z+4} \), simplifying fractions ensures that the answer is in its most understandable form.

Consider the fraction \( \frac{1}{-5 + 4} \). After performing the arithmetic operation in the denominator, it becomes \( \frac{1}{-1} \). Here, simplifying is straightforward: dividing 1 by -1 simply flips the sign, giving \(-1\).

When simplifying fractions:

• Perform all arithmetic operations within the numerator and denominator first.
• If possible, reduce the fraction to its simplest form. For instance, \( \frac{4}{2} \) simplifies to \( 2 \).
• Simplify negative signs: \( \frac{1}{-1} \) becomes \(-1\) to keep the expression cleaner and easier to interpret.

This skill is not only essential in function evaluation but also in solving algebraic equations and various mathematical computations. Understanding fraction simplification leads to more accurate and elegant solutions.