Math: What a Function Is and Does

This is Unit Ten of ACT Preparation: what a function is and does. Unit nine treated the line as a rate and a starting value; this unit generalizes the line into any rule that turns an input into one output. One question runs through it: which number is the input, and which is the output? By the end you will be able to evaluate and solve with function notation. To find a domain and range. To find an average rate of change. To compose and transform functions. And to find and check an inverse.

Six entries on what a function is and how its notation works.

One output for each input.

ACT 0901 What a Function Is

A relation is a set of ordered pairs; a function gives each input exactly one output. The pairs one, four; two, four; and three, seven form a function, but one, four and one, five do not. Two inputs may share an output; one input may not have two.

Function notation.

ACT 0902 Function Notation

In f of x, f names the function and x is the input; f of a is the output when the input is a. For f of x equals x squared minus 3x, f of four is four, and f of negative two is ten. F of x is not f times x, and a negative input goes in parentheses.

What a function value means.

ACT 0903 Function Values in Context

In context, f of a equals b means the input a produces the output b, in the situation's units. If C of h is a repair bill after h hours, C of three equals 135 means three hours cost $135. The number inside the parentheses is the input.

Evaluating and solving, step by step.

ACT 0904 Evaluating and Solving With Function Notation

To evaluate, substitute the input and simplify; to solve, set the formula equal to the output and solve for x, which may give several answers. For f of x equals 2x squared minus five, f of negative three is thirteen, and f of x equals twenty-seven gives four or negative four. If g of two and g of five both equal nine, solving g of x equals nine gives both. Solving can return several inputs; evaluating always returns one output.

Telling a function from its graph.

ACT 0905 The Vertical Line Test

A graph shows a function exactly when no vertical line crosses it more than once. A parabola opening up passes; a circle fails. This test checks for a function, not for one-to-one.

One-to-one functions.

ACT 0906 One-to-One and the Horizontal Line Test

A function is one-to-one when each output comes from only one input, and its graph passes the horizontal line test. Two x plus one is one-to-one; x squared is not, since three and negative three both give nine. Only one-to-one functions have inverses.

Five entries on domain, range, and functions built in pieces.

The allowed inputs.

ACT 0907 Domain

The domain is the set of allowed inputs. Exclude inputs that divide by zero or take an even root of a negative, and any the situation rules out: one over x minus four excludes four. In context the domain shrinks: a number of tickets cannot be negative.

The outputs produced.

ACT 0908 Range

The range is the set of outputs the function actually produces; on a graph, read the domain across and the range up. X squared plus one has all real numbers as its domain and outputs of at least one as its range, because squaring never produces negatives.

Finding a domain, step by step.

ACT 0909 Finding a Domain

Start from all real numbers, find each restriction, solve it, and combine. For the square root of six minus 2x over x plus one, the root needs x at most three and the denominator needs x not equal to negative one. Solve the root's inequality carefully: dividing by negative two reverses the sign, so x is at most three, not at least.

Functions built in pieces.

ACT 0910 Piecewise Functions

A piecewise function uses different formulas on different parts of its domain; evaluate it by first finding which piece holds the input. With x plus two below zero and x squared from zero up, f of negative three is negative one and f of three is nine. At the boundary, the included sign decides.

A piecewise function, step by step.

ACT 0911 Evaluating a Piecewise Function

Read each piece's interval and its endpoints, locate the input in exactly one, and apply only that formula. A garage charges $5 for up to two hours, $3 for each extra hour up to six, and a flat $20 above six: two hours cost $5, five hours $14, seven hours $20. A boundary input belongs to the piece that includes it, so six hours costs $17, not $20.

Three entries on rates of change and on where a function rises and falls.

Average rate of change.

ACT 0912 Average Rate of Change

The average rate of change from a to b is the change in output over the change in input: the slope of the line joining two points of the graph. For x squared from one to four, fifteen over three is five. For a curve, it depends on the interval.

Rising, falling, peaks and valleys.

ACT 0913 Increasing, Decreasing, Maximum and Minimum

A function increases where its graph rises left to right and decreases where it falls; a local maximum is a peak and a local minimum a valley. Negative the quantity x minus two, squared, plus five rises until two, reaches a maximum of five, then falls. Report the maximum as a value and its location as an input.

Rates from a table, step by step.

ACT 0914 Average Rate of Change From a Table

Read the outputs at the two inputs, subtract both in the same order, divide, and attach units. A town of forty thousand in 2010, forty-six in 2014 and fifty-eight in 2020 grew 1.5 thousand a year, then two thousand a year, and 1.8 thousand a year over the whole span. That is not the average of the two rates, 1.75, because the intervals differ in length.

Nine entries on composing functions and transforming their graphs.

Combining functions.

ACT 0915 Composition

The composition f of g of x applies g first, then f to the result, and order matters. With f adding three and g doubling, f of g of five is thirteen, but g of f of five is sixteen. Work from the inside out.

Evaluating a composition, step by step.

ACT 0916 Evaluating a Composition

Evaluate the inner function, use its output as the outer function's input, and evaluate the outer function. From the table, g of two is one, and f of one is four. As formulas, with f squaring and g subtracting four, f of g of x is x minus four, squared, while g of f of x is x squared minus four. Look up the inner output in the outer function's row.

Moving a graph up and down.

ACT 0917 Vertical Shifts

Adding a constant outside the function shifts the graph up by that amount; subtracting shifts it down. X squared plus three is the parabola moved up three. The range moves, and the domain does not.

Moving a graph sideways.

ACT 0918 Horizontal Shifts

Replacing x with x minus h shifts the graph right by h, and x plus h shifts it left. X minus four, squared, has its vertex at four, zero. The sign runs opposite to intuition: ask what input makes the inside zero.

Reflections.

ACT 0919 Reflections

Negative f of x reflects the graph over the x-axis, turning it upside down; f of negative x reflects it over the y-axis. The position of the minus sign decides the axis: outside affects outputs, inside affects inputs.

Stretches and compressions.

ACT 0920 Stretches and Compressions

Multiplying the function by a number above one stretches the graph vertically, and by a number between zero and one compresses it: 3x squared is narrower than x squared. Inside changes work in reverse: f of 2x squeezes the graph toward the y-axis by half.

The V-shaped graph.

ACT 0921 The Absolute Value Function

The absolute value function makes a V with its vertex at the origin, and in a times the absolute value of x minus h, plus k, the vertex moves to h, k. The absolute value of x plus one, minus two, has its vertex at negative one, negative two. A negative a turns the V upside down.

Writing a transformed function, step by step.

ACT 0922 Writing a Transformed Function

Start from the base function, apply the inside changes for horizontal moves, then the outside changes: stretch or reflect, then shift. Reflecting the square root over the x-axis, shifting left three and up two gives negative root x plus three, plus two, and the origin moves to negative three, two. Reflect before shifting: shifting up first would put the point at negative two.

Matching a graph to its equation, step by step.

ACT 0923 Matching a Graph to Its Equation

Identify the base shape, read the key point for h and k, check direction and steepness, and confirm with another point. A V opening down with vertex two, three, through three, one, is negative two times the absolute value of x minus two, plus three. A vertex at two means x minus two inside the bars, not x plus two.

Three entries on inverse functions.

Undoing a function.

ACT 0924 Inverse Functions

The inverse of a one-to-one function undoes it: if f sends a to b, f inverse sends b back to a. Two x plus three sends four to eleven, and its inverse sends eleven back to four. F inverse is not one over f; the negative one marks the inverse, not a reciprocal.

Finding an inverse.

ACT 0925 Finding an Inverse

To find an inverse, write y equals f of x, swap x and y, and solve for y: from y equals 2x plus three, the inverse is x minus three, over two. The inverse's graph reflects f's over the line y equals x, and the domain and range swap.

Finding and checking an inverse, step by step.

ACT 0926 Finding and Checking an Inverse

Confirm the function is one-to-one, swap x and y, solve, and check by composing both ways. For x plus five, over three, the inverse is 3x minus five, and both compositions simplify to x. A function that is not one-to-one, such as x squared, has no inverse until its domain is restricted, to x at least zero, where the inverse is the square root.

Look at what you can now do. You can tell a function from a relation, and evaluate or solve with its notation. You can find a domain from its restrictions, read a range, and pick the right piece of a piecewise function. You can find an average rate of change, with units, over any interval. You can compose from the inside out and move, flip and stretch a graph. And you can find an inverse and prove it by composing both ways. In every case, the first move was naming the input and the output. Unit eleven turns to quadratics and other curves.