91Ó°ÊÓ

Function addition is when you combine two functions by adding their outputs. It's like adding two numbers together. For example, if you have two functions, say \(f(x)\) and \(h(x)\), and you are asked to find \((f+h)(x)\), you simply do:

• Add the expressions for the functions.
• Combine like terms if necessary.

Let's consider the functions given in the exercise: \(f(x) = x^2 - 9\) and \(h(x) = x - 3\). To find \((f + h)(x)\), we add the two functions:

\((f + h)(x) = f(x) + h(x)\)

Substituting the given functions, we get:

\((f + h)(x) = (x^2 - 9) + (x - 3)\)

So, combining like terms, it becomes:

\((f + h)(x) = x^2 + x - 12\). Now you can substitute any value for \(x\).

Function substitution, also known as evaluating a function, means plugging a specific value for \(x\) into the function and finding the result.

For example, to evaluate \(f(x)\) at \(x = -2\), you would perform:

\(f(-2) = (-2)^2 - 9\)

This simplifies to \(4 - 9\), which equals \(-5\).

Similarly for \(h(x)\), substituting \(x=-2\) we get:

\(h(-2) = -2 - 3 = -5\)

So, function substitution involves replacing \(x\) with a specific value and simplifying.

In general, evaluating functions is the process of finding the value of a function for a specific value of \(x\). Here’s how you can do it step-by-step:

• Identify the function you need to evaluate, such as \(f(x)\) or \(h(x)\).
• Replace \(x\) in the function with the given value.
• Simplify the expression to find the result.

In the exercise, we need to evaluate \((f+h)(-2)\). We already calculated:

\(f(-2) = -5\) and \(h(-2) = -5\)

So to find \((f + h)(-2)\), we add these results together:

\((f + h)(-2) = f(-2) + h(-2) = -5 + (-5) = -10\)

This shows that evaluating functions can help you find the output from specific inputs, such as \(-2\) in this exercise.