Precalculus Functions, Domain and Range Lesson (Free PDF)

Q: What does f(x) mean?

The symbol f(x) means the output of the function f when the input is x. It is not f times x. For f(x) = 3x - 2, f(4) = 3(4) - 2 = 10.

Q: How do you find the domain of a rational function?

Set the denominator equal to 0 and solve. Those x-values are excluded. For 1/(x - 4), the domain is all real numbers except 4.

Q: What is the difference between domain and range?

The domain lists inputs (x-values) and the range lists outputs (y-values). Domain is read left to right on a graph and range is read bottom to top.

Q: Why does a circle fail the vertical line test?

Most vertical lines cross a circle twice, so one x-value would need two y-values. A function allows only one output per input, so a circle is not a function.

Q: When do you use a bracket and when a parenthesis?

Use a bracket when the endpoint is included, as in [2, 5]. Use a parenthesis when the endpoint is excluded or the interval goes to infinity, as in (3, infinity).

Precalculus math lesson: Functions, Domain and Range – Maths PDF

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Sommaire

This precalculus functions lesson for Precalculus covers functions, domain and range (Common Core F-IF.A.1, F-IF.A.2, F-IF.B.5), with a free PDF to print.

A function assigns each input exactly one output, written f(x). The domain is the set of allowed inputs and the range is the set of outputs produced. To find a domain, remove any x that makes a denominator zero or puts a negative number under an even root, then write the result in interval notation.

Precalculus · Common Core State Standards: F-IF.A.1, F-IF.A.2, F-IF.B.5

Key facts in this precalculus functions lesson

Key facts

A relation is a function only if each input has exactly one output (vertical line test).
Domain: allowed inputs. Range: outputs that actually occur.
Exclude values that make a denominator 0 or an even-root radicand negative.
Interval notation uses brackets for included endpoints and parentheses for excluded ones.
A piecewise function uses a different rule on each part of its domain.

What is a function?

A relation is a function when every input has exactly one output. Mathematicians write the output of the function \(f\) at the input \(x\) as \(f(x)\). To evaluate, replace every \(x\) with the input. For \(f(x)=2x+1\), \(f(3)=2(3)+1=7\).

A function is a rule: each input goes in once and exactly one output comes out.

Domain and range

The domain is the set of all allowed inputs. The range is the set of all outputs the function actually produces.

How do you find the domain of a function?

Start with all real numbers, then remove inputs that break a rule of arithmetic. Two rules cover most of Precalculus:

You cannot divide by zero, so exclude any \(x\) that makes a denominator \(0\).
You cannot take an even root of a negative number, so require the radicand to be \(\ge 0\).

Example: For \(f(x)=\sqrt{x-1}\) we need \(x-1\ge 0\), so \(x\ge 1\). The domain is \([1,\infty)\). The smallest output is \(f(1)=0\), so the range is \([0,\infty)\).

The graph of \(f(x)=\sqrt{x-1}\) begins at \((1,0)\): domain \([1,\infty)\), range \([0,\infty)\).

How do you read domain and range from a graph?

The domain is the set of \(x\)-values the graph covers from left to right. The range is the set of \(y\)-values it covers from bottom to top. A closed dot means the endpoint is included, so use a bracket. An open dot or an arrow past the edge means use a parenthesis.

Set	Interval notation	Inequality
All numbers from 2 to 5, including both	\([2,5]\)	\(2\le x\le 5\)
Greater than 3	\((3,\infty)\)	\(x>3\)
All reals except 4	\((-\infty,4)\cup(4,\infty)\)	\(x\neq 4\)

How do you test whether a graph is a function?

Use the vertical line test: if any vertical line meets the graph in more than one point, one input has two outputs, so the graph is not a function.

The vertical line at x = 1.5 hits the circle twice, so a circle is not the graph of a function.

What is a piecewise function?

A piecewise function uses different rules on different parts of its domain. Check which condition the input satisfies, then use only that rule. The graph below uses \(y=x+4\) when \(x< 1\) and \(y=x^{2}\) when \(x\ge 1\).

An open circle marks an endpoint that is not included; a closed point marks one that is.

Precalculus functions lesson: questions and answers

What does f(x) mean?

The symbol f(x) means the output of the function f when the input is x. It is not f times x. For f(x) = 3x – 2, f(4) = 3(4) – 2 = 10.

How do you find the domain of a rational function?

Set the denominator equal to 0 and solve. Those x-values are excluded. For 1/(x – 4), the domain is all real numbers except 4.

What is the difference between domain and range?

The domain lists inputs (x-values) and the range lists outputs (y-values). Domain is read left to right on a graph and range is read bottom to top.

Why does a circle fail the vertical line test?

Most vertical lines cross a circle twice, so one x-value would need two y-values. A function allows only one output per input, so a circle is not a function.

When do you use a bracket and when a parenthesis?

Use a bracket when the endpoint is included, as in [2, 5]. Use a parenthesis when the endpoint is excluded or the interval goes to infinity, as in (3, infinity).

How to use this precalculus functions lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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