Functions
A function is a relationship between two variables. Usually written as $f(x)$ or $g(x)$, functions are just equations (like $y = x– 4$). For example, $f(x)= x - 4$ means that for every value of $x$, $f(x)$ is $x - 4$ or $y=x-4$.
Functions tend to scare students. For many students, as soon as they see $f(x)$, they assume that they cannot do a problem. But functions just show how two variables are related to each other.
Try to think of functions as equations. $f(x)$ is the same as $y$, which is the output or range of the equation. $x$ (or whatever is in the parentheses) is $x$, or the input or domain of the equation.
Very simply, $f(x)$ is exactly the same as $y$, except that it also acts like a little container -- and that container holds your $x$ for you. It's $y$ with pocket that holds your value. It's handy. Don't be scared by it!
Whether you’re solving or creating a function, just think of the function as a regular equation.
If you’re given a function: $f(3) = x +7$ Substitute the number in parentheses for $x$ in the equation to solve. $f(3) = 3 +7$ $f(3) = 10$
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If you’re given a table, create the function by finding how the numbers are related. What do you do to x to get $f(x)$? $f(x) = 3x + 1$ Try it: $3(1)+1=4$ and $3(2)+1=7$ |
When you're dealing with a function, remember that the number inside the parentheses is the $x$ in your equation (or table of values) and the $f(x)$ or $g(m)$ term is the $y$.
Example: The table gives the values of function $f(m)$ for several values of $m$. If the graph of $f(m)$ is a line, which of the following equations defines $f(m)$?
| $m$ | 1 | 2 | 3 | 4 |
| $f(m)$ | 4 | 8 | 12 | 16 |
1. First, figure out the relationship between $m$ and $f(m)$. If it helps, cross out $m$ and write $x$ and cross out $f(m)$ and write $y$.
| How do you get $f(m)$ if you start with $m$? For each value of $m$, you multiply by $4$ to get4 f(m)$, so the rule or “function” is multiply by 4. Try it for each value of m: $$\eqalign{m&=1\qquad4\times1&&4\\m&=2\qquad 4\times2&&=8\\m&=3\qquad 4\times3&&=12\\m&=4\qquad 4\times4&&=16}$$ The rule works for each value of $m$ which means that for every value of $m$, $f(m)=4 \times m$.
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| 2. Write that relationship as an equation | $f(m) = 4m$ |
Overall, functions are just equations and the $f(x)$ notations just allows the $x=?$ information to be embedded in the $y$. Just figure out patterns, and plug in numbers, and you'll get them right! Remember, bottom line: $x=x$ and $f(x)=y$.
Practice Problems:
Functions
$f(x)=3x+7$
Find the value of:
- $f(2)$
- $f(3)$
- $f(5)$
- $f(10)$
$f(2)=7$
- Which of the following could be f(x)?
- $f(x)=x+2$
- $f(x)=x^2-1$
- $f(x)=3x-1$
- $f(x)=x^2-2$
- $f(x)=x^3-1$
$f(x)=2x-4$
$g(x)=\dfrac{x}{2}+3$
Find the value of:
- $f(4)$
- $g(3)$
- $f(g(2))$
- $g(f(1))$
- $f(g(f(2)))$
Answer Key:
Test Prep Practice
Functions Test Prep
1.
$x$ 1 2 3 4 $f(x)$ 1 -2 -5 -8 The table above shows a relationship between $x$ and $f(x)$. Which of the following equations describe that relationship?
(A) $f(x)=-2x+6$
(B) $f(x)=4x-10$
(C) $f(x)=x-8$
(D) $f(x)=-2x+1$
(E) $f(x)=-3x+4$
2. For which of the following functions is $g(-5)>g(5)$?
(A) $g(x) = x^3$
(B) $g(x) = x^2-x$
(C) $g(x) = 4x -9$
(D) $g(x) = |x^3|$
(E) $g(x) = (-2x + x)^2$
3. The figure above shows the graph of the function $h(x)$. Which of the following $h(x)$ values has only one $x$?
(A) $h(x)=-5$
(B) $h(x)=-1$
(C) $h(x)=1$
(D) $h(x)=4$
4. If $f(x)= 8x^2-4x+7$, find $f(-3)$.
Grid in your answers
Answer Key: