Evaluating Functions

 

By Robert O

What is a function?

A function is an expression that accepts an input and gives a corresponding output. Think of a function as a bread-making factory. It takes in wheat flour plus other ingredients and gives out bread. Functions define actions that take place to turn input into output.

How do we write functions?

Before we get to the evaluation of function, we need to know how to represent functions. The most famous notation is f(x), read as f of x. However, you can you other notations such as h(x), g(x), or any other letter. The variable x can also change. For example, f(r), read as f of r.

How to evaluate functions?

It is a straightforward process to evaluate functions. It simply means replacing the variable in it with a value. We will demonstrate that using a few examples. Also, note that you can evaluate functions using other expressions, as we will see soon.

Example 1:

Evaluate f(x) = x+8 at x=2

Solution

Substitute x with 2 in the above function.

    \[\mathrm{f}(2)=2+8=10\]

Example 2:

Given

    \[f(x)=x^{2}+4 x-2\]

find f(1).

Solution

Substitute x=1 in the above function.

    \[\mathrm{f}(1)=1+4-2=3\]

Example 3:

Given

    \[f(x)=3 x^{2}+4 x-2\]

evaluate f(n+2).

Solution

In this case, we have an expression instead of a value. The evaluation process does not change. Just replace x with n+2.

    \[f(n+1)=(n+1)^{2}+4(n+1)-2\]

    \[\mathrm{f}(\mathrm{n}+1)=\mathrm{n}^{2}+2 n+1+4 \mathrm{n}+4-2\]

    \[\mathrm{f}(\mathrm{n}+1)=\mathrm{n}^{2}+6 n+3\]

Example 4:

Given

    \[\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-2 x-5 \text { and } \mathrm{f}(\mathrm{z})=10\]

find the value of z.

Solution

For a nested function like this example, you need to first evaluate the outer function and then the inner one.

    \[\mathrm{f}(\mathrm{k})=\mathrm{k}^{2}-2 k-5\]

We now set the equation equal to 10 and evaluate the value of k.

    \[10=\mathrm{k}^{2}-2 k-5\]

This is a quadratic equation:

    \[\mathrm{k}^{2}-2 k-15=0\]

Using the quadratic formula: 

    \[x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\]

    \[x=\frac{2 \pm \sqrt{4+60}}{2}\]

    \[x=5 \text { or }-3\]

Example 5:

What is the value of k if

    \[f(x)=3 x^{2}+k x-1\]

and f(2)= 6?

Solution

We start by substituting the value of x in the function.

    \[\mathrm{f}(2)=3(2)^{2}+2 \mathrm{k}-1\]

    \[\mathrm{f}(2)=11+2 \mathrm{k}\]

We know that f(2) = 6. Equating this to the previous equation we get:

    \[6=11+2 \mathrm{k}\]

    \[k=-2.5\]

Remarks

You can evaluate any function no matter how complex it is. You only need to be accurate and use parenthesis where applicable to avoid mistakes.

About the Author

This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.