Functions

Functions

A function is a rule that takes an input and produces exactly one output.

Function notation is written as:

This is read as "\(f\) of \(x\)." The expression \(f(x)\) represents the output produced by the function when the input is \(x\).

For example:

To calculate \(f(4)\), substitute 4 wherever \(x\) appears:

A function must assign exactly one output to each input. Different inputs can produce the same output. For example:

Then:

and:

Both inputs produce the same output, which is allowed.

4.1 Domain and range

The domain is the set of allowed inputs to a function.

The range is the set of outputs the function can produce.

Consider:

Any real number can be used as an input, so the domain is all real numbers.

Because \(x^2\) cannot be negative:

Therefore:

The smallest possible output is 2. The range is:

Now consider:

The input cannot be zero because division by zero is undefined.

Its domain is all real numbers except zero. Its range is also all real numbers except zero.

For:

the denominator cannot be zero:

Therefore, \(x\ne3\).

4.2 Linear functions

A linear function has the general form:

where:

• \(m\) is the slope.
• \(c\) is the \(y\)-intercept.

The slope describes how quickly the output changes as the input changes.

The \(y\)-intercept is the output when \(x=0\).

For example:

Its slope is 2, and its \(y\)-intercept is 3.

When \(x=0\):

So the graph crosses the \(y\)-axis at \((0,3)\).

4.3 Plotting a linear function

Consider:

Some input-output pairs are:

These give the coordinates:

Plotting these points and connecting them produces a straight line.

4.4 Calculating slope from two points

Given two points:

the slope is:

For example, using the points \((-2,0)\) and \((0,2)\):

The slope is 1.

4.5 Positive, negative, and zero slopes

A positive slope means the line increases as \(x\) increases.

A negative slope means the line decreases as \(x\) increases.

A zero slope means the function is constant: its output does not change as the input changes.

For example:

has slope 0.

A function such as:

has slope \(-2\) and \(y\)-intercept 0.

4.6 The x-intercept and y-intercept

The \(y\)-intercept is found by setting \(x=0\).

The \(x\)-intercept is found by setting \(y=0\), or equivalently solving \(f(x)=0\).

For:

The \(y\)-intercept is:

To find the \(x\)-intercept:

So the \(x\)-intercept is:

4.7 Parallel and perpendicular lines

Two non-vertical lines are parallel when they have the same slope.

For example:

and:

Both have slope 3, so they are parallel.

Two non-vertical lines are perpendicular when the product of their slopes is \(-1\):

For example, a line with slope 2 is perpendicular to a line with slope:

A horizontal line has slope 0. A vertical line has an undefined slope.

---