Functions

​​In a nutshell

A function maps a set of input values, or the function's domain, onto a set of output values, or the range of the function. The notation $$f(x)$$​ represents a function of $$x$$. The roots of a function are the values of $$x$$ for which $$f(x)=0$$.​

Definitions

Functions

You can think of a function as a machine that takes a set of inputs (domain) to produce a set of outputs (range). $$f(x)$$ represents a function of $$x$$.​

The diagram below shows how the function $$f(x)=x+2$$ maps three values from its domain to values in its range.

The function $$f(x)=x+2$$ is an example of a one-to-one function, where one value in its domain will put out only one value in its range.

Some functions are many-to-one, where the function has multiple values in its domain that all map to the same value in its range.

For example, consider the function $$f(x)=x^2$$. From the diagram you can see that both values $$-2$$ and $$2$$ map to the same value ($$4$$).

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Example 1

The functions $$f$$ and $$g$$ are given by $$f(x)=4x-3$$ and $$g(x)=x^2+1$$, $$x\in \reals$$. Find $$f(4)$$ and $$g(7)$$. What is the value for which $$f(x)=g(x)$$?

Note: $$x \in\reals$$ means that $$x$$ is any real number.

Substitute for $$x$$​ for both functions:

$$\begin{aligned}f(4)&=4(4)-3=\underline{13}\\g(7)&=7^2+1=\underline{50}\end{aligned}$$​​

Set $$f(x)=g(x)$$​ and solve for $$x$$:

​$$\begin{aligned}f(x)&=g(x)\\4x-3&=x^2+1\\x^2-4x+4&=0\\(x-2)^2&=0\\x&=\underline{2}\end{aligned}$$​

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Example 2

Find all the roots of the functions $$f(x)=3x-15$$ and $$g(x)=121-x^2$$.

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A root is a value of $$x$$ for which the function is zero. So, you need to find $$x$$ for $$f(x)=0$$ and $$g(x)=0$$.

$$\begin{aligned}f(x)=3x-15&=0\\3x&=15\\x&=\underline{5}\end{aligned}$$​​

$$\begin{aligned}g(x)=121-x^2&=0\\x^2&=121\\x&=\underline{\pm11}\end{aligned}$$​​

Note: $$121$$ has more than one square root. You must include both $$11$$ and $$-11$$. You can write this as '$$\pm11$$'.

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