Function problems

Think of a function as a machine that takes an input and produces an output. For example, if the machine is described by the expression $x+7$ and you put in the input $2$ (meaning you use $x=2$), the output is $9$ because $2+7=9$. This is the main idea behind function problems. The part that usually causes confusion is the notation.

The function described above is represented as:

$f(x)=x+7$

This reads: “$f$ of $x$ equals $x+7$.” The input is whatever is inside the parentheses (here, $x$). The function rule tells you what to do with that input: take the value of $x$ and compute $x+7$ to get the output.

Notice that there is no variable named $f$ in this equation. The letter $f$ is just the name of the function.

Try solving this simple function:

Given: $f(x)=4x+2$

What is $f(3)$?

Do you know the answer?

Simple enough, right?

Nested functions

Sometimes you’ll see one function used inside another function.

Given: $f(x)=x+3$ and $g(x)=3x$

What is $f(g(3))$?

The notation can look confusing at first, but the process is the same as before. Work from the inside out:

• Evaluate the inner function $g(3)$.
• Use that result as the input to the outer function $f(\cdot )$.

Try solving this and then check your work below.

Multiple input functions

Sometimes a single function can have multiple inputs. For example, $f(x,y)=2x-y$ is a function with two inputs. To find $f(2,1)$, replace $x$ with $2$ and $y$ with $1$.

$$\begin{array}{rl}f(x,y)& =2x-y\\ f(2,1)& =2(2)-1\\ f(2,1)& =4-1\\ f(2,1)& =3\end{array}$$

Symbolic functions

Sometimes a function is written using an unusual symbol instead of a letter like $f$ or $g$. For example, you might see a question like this:

What is the value of $3\star$, given that $x\star =x-2.5$?

Have a guess?

To solve these, treat the symbol as the name of a function. Here, $x\star$ plays the same role as $f(x)$. So you can rewrite the rule as a standard function and evaluate it.

$$\begin{array}{rl}x\star & =x-2.5\\ f(x)& =x-2.5\\ f(3)& =3-2.5\\ f(3)& =0.5\end{array}$$

These symbols might be hearts, boxes, stars, diamonds, or something else that looks out of place. When you rewrite the rule using standard function notation, the problem becomes much easier to read.

Bringing it all together: question walkthrough video

Here’s a video going through one of our practice questions to demonstrate these ideas in action: