A. Characteristics of Functions · Functions and Equivalent Expressions

Function Notation: Naming, Evaluating and Reading Functions

A1.2About 70 minutes

Learning goals

I can read and write function notation such as f(x) = 2x² + 3x − 1.
I can evaluate a function for numbers and for algebraic expressions.
I can find f(a) and solve f(x) = k from an equation, a table or a graph.
I can write a linear or quadratic function in function notation from a table or a graph.

Words to know

function notation
A way of writing a function that names it and shows its input, such as f(x) = 3x + 2. f(x) is the output.

evaluate
Find the value of a function for a given input by substituting.

f(a)
The output of f when the input is a; the point (a, f(a)) is on the graph.

argument
Another name for the input inside the brackets of f( ).

constant function
A function whose output is the same number for every input, such as f(x) = 7.

A name for every rule

In Grade 9 and 10 you wrote equations like y = 2x² + 3x − 1. That works when there is one relation on the page. But a physics problem might have a height rule, a speed rule and a cost rule all at once. Calling every output "y" gets confusing fast.

Function notation gives each function a name and shows the input right beside it. Instead of y = 2x² + 3x − 1 we write

f(x) = 2x² + 3x − 1

read "f of x equals 2x² + 3x − 1".

What the symbols mean

f is the name of the function. Other common names are g, h, or a letter that fits the story, such as h for height or C for cost.
x inside the brackets is the input.
f(x) is the output: the value of the function at x. So y = f(x), and the point (x, f(x)) is on the graph.

To evaluate f(3), replace every x in the rule with 3:

f(3) = 2(3)² + 3(3) − 1 = 18 + 9 − 1 = 26.

The statement f(3) = 26 means the point (3, 26) is on the graph of f.

Each part of f(3) = 26 has a job. The brackets hold the input; they are not multiplication.

Worked example

Example 1: Evaluating a quadratic function

Given f(x) = 2x² + 3x − 1, evaluate (a) f(−2) (b) f(0) (c) f(1/2).

Put each input in brackets before substituting. This protects the sign when you square a negative.
(a) f(−2) = 2(−2)² + 3(−2) − 1 = 2(4) − 6 − 1 = 1.
(b) f(0) = 2(0)² + 3(0) − 1 = −1. (f(0) is always the y-intercept.)
(c) f(1/2) = 2(1/2)² + 3(1/2) − 1 = 2(1/4) + 3/2 − 1 = 1/2 + 3/2 − 1 = 1.

f(−2) = 1, f(0) = −1, f(1/2) = 1. So (−2, 1), (0, −1) and (1/2, 1) are on the graph.

Brackets save marks

Without brackets, −2² on a calculator gives −4, because the calculator squares 2 first and then applies the negative. With brackets, (−2)² = 4. Always type and write negative inputs in brackets.

Algebraic inputs

The input does not have to be a number. Whatever is in the brackets replaces every x.

Compare these carefully. They look alike but mean different things:

f(x + 1): add 1 to the input, then apply f.
f(x) + 1: apply f, then add 1 to the output.
f(2x): double the input first.
2f(x): double the output.

Worked example

Example 2: Substituting expressions

Given g(x) = −x² + 4x, find and simplify (a) g(a) (b) g(x + 1) (c) g(x) + 1 (d) g(2x).

(a) Replace x with a: g(a) = −a² + 4a.
(b) Replace x with (x + 1): g(x + 1) = −(x + 1)² + 4(x + 1) = −(x² + 2x + 1) + 4x + 4 = −x² − 2x − 1 + 4x + 4 = −x² + 2x + 3.
(c) Take the whole output and add 1: g(x) + 1 = −x² + 4x + 1.
(d) Replace x with (2x): g(2x) = −(2x)² + 4(2x) = −4x² + 8x. Note (2x)² = 4x², not 2x².

g(a) = −a² + 4a; g(x + 1) = −x² + 2x + 3; g(x) + 1 = −x² + 4x + 1; g(2x) = −4x² + 8x. Parts (b) and (c) are different.

Going backwards: solving f(x) = k

Sometimes you know the output and want the input. "Find x so that f(x) = 11" means: set the rule equal to 11 and solve. A quadratic may give two inputs (remember: many-to-one is allowed for a function).

Worked example

Example 3: Solving for the input

(a) f(x) = 3x − 4. Find x so that f(x) = 11. (b) q(x) = x² − 6. Find x so that q(x) = 10.

(a) 3x − 4 = 11, so 3x = 15 and x = 5. Check: f(5) = 15 − 4 = 11. ✓
(b) x² − 6 = 10, so x² = 16 and x = 4 or x = −4. Check: q(4) = 16 − 6 = 10 and q(−4) = 16 − 6 = 10. ✓

(a) x = 5. (b) x = ±4: two inputs give the output 10.

Reading function notation from a graph

To find f(a): go to x = a on the x-axis, move up or down to the graph, and read the y-value.
To solve f(x) = k: draw the horizontal line y = k and read the x-values where it meets the graph.

The graph of f(x) = −(x − 1)² + 4. Reading at x = 3 gives f(3) = 0. The line y = 3 meets the graph twice, so f(x) = 3 when x = 0 or x = 2.

Worked example

Example 4: Writing a function from a table

Write each table as a function in function notation.

Table A: x = 0, 1, 2, 3 gives y = 5, 8, 11, 14. Table B: x = 0, 1, 2, 3, 4 gives y = 3, 0, −1, 0, 3.

Table A. First differences are 3, 3, 3: constant, so the relation is linear with slope 3. The y-intercept is f(0) = 5. So f(x) = 3x + 5.
Table B. First differences: −3, −1, 1, 3. Second differences: 2, 2, 2: constant, so the relation is quadratic (Grade 10).
The table is symmetric about x = 2, and the lowest value is −1 at x = 2. So the vertex is (2, −1) and g(x) = a(x − 2)² − 1.
Use the point (0, 3): 3 = a(0 − 2)² − 1, so 4a = 4 and a = 1. So g(x) = (x − 2)² − 1.
Check another point: g(1) = (1 − 2)² − 1 = 1 − 1 = 0. ✓

f(x) = 3x + 5 and g(x) = (x − 2)² − 1.

Worked example

Example 5: Function notation in a story

A ball is thrown upward from a deck. Its height in metres after t seconds is h(t) = −5t² + 20t + 1. Find h(0), h(2) and h(4), and explain what each means.

h(0) = −5(0)² + 20(0) + 1 = 1. The ball starts 1 m above the ground (it leaves the thrower's hand at that height).
h(2) = −5(4) + 20(2) + 1 = −20 + 40 + 1 = 21. After 2 s, the ball is 21 m high.
h(4) = −5(16) + 20(4) + 1 = −80 + 80 + 1 = 1. After 4 s the ball is back down at 1 m, on its way down.

h(0) = 1, h(2) = 21, h(4) = 1. The notation tells you the time (input) and the height (output) in one symbol.

Just for fun

Why are functions so often called f, g and h? The first one is f for function. When mathematicians needed a second one, they just took the next letter of the alphabet, and then the next. Not every naming decision in math is deep.

Hands-on activity

Build a function machine

You need: an empty cereal box or envelope, slips of paper, a pencil, and a partner (a grown-up works well).

Your partner secretly writes a rule on a card, such as f(x) = 2x − 1 or f(x) = x² + 1, and keeps it inside the box.
You feed in inputs on slips of paper: 0, 1, 2, 3 and −1. Your partner returns each slip with the output written on it, in the form f(2) = 3.
Record the results in a table. Use first and second differences to decide whether the rule is linear or quadratic.
Write the rule in function notation and test it on a new input before opening the box.
Swap roles.

What to look for: for f(x) = 2x − 1 the outputs are −1, 1, 3, 5 and −3 (first differences 2). For f(x) = x² + 1 they are 1, 2, 5, 10 and 2 (second differences 2). Your partner should record outputs as f(input) = output.

Your turn

If f(x) = 5x − 7, find f(3) and f(−2).

If g(x) = x² − 3x + 2, find g(0), g(1) and g(−1).

If f(x) = 4 − x², find f(1/2).

f(5) = 12. Which point is on the graph of f?

If h(x) = 2x + 9, solve h(x) = 1.

If k(x) = 3x², find k(x − 1) and simplify.

Write y = −x² + 6 in function notation using the name p.

More practice: warm-up

How do you read f(x)?

If f(x) = x + 10, find f(4).

If g(x) = 2x², find g(−3).

If f(0) = 6, what is the y-intercept of the graph of f?

If m(x) = 7, find m(100).

More practice: on your own

f(x) = 2x² − x + 4. Find f(−3) and f(2a).

f(x) = x² + 2x. Find f(x + 3) − f(x) and simplify.

g(x) = x² − 4x. Solve g(x) = 5.

A table has x = 1, 2, 3, 4 and y = −1, 1, 3, 5. Write it as a function f(x).

The cost in dollars to rent a canoe for h hours is C(h) = 12h + 15. Find C(3) and explain its meaning. Then solve C(h) = 75.

Challenge

f(x) = 3x − 2. Find a so that f(a) = f(2a) − 10.

A quadratic function has f(0) = 5, f(1) = 2 and f(2) = 1. Find f(x) in the form ax² + bx + c.

For f(x) = x², show that f(a + b) ≠ f(a) + f(b) in general, and find when they are equal.

Check your understanding

Quiz

If f(x) = 3x² − 2x, what is f(−1)?

What does g(4) = −3 tell you?

If f(x) = x² + 1, which expression equals f(x + 2)?

If h(x) = 2x − 6, for which x is h(x) = 0?

A function has f(x) = 4 when x = −1 and when x = 5. What kind of mapping is this, and is f a function?

A table for a linear function has f(0) = −2 and f(1) = 1. What is f(x)?

For parents and teachers

Big idea: Expectation A1.2 asks students to write linear and quadratic functions in function notation (from equations, tables or graphs) and to substitute into and evaluate them.

How to help: Ask your teen to "translate" between the symbols and plain words: "f(2) = 7 means when the input is 2, the output is 7, so (2, 7) is on the graph." Have them evaluate a function at a negative number aloud and listen for the brackets.

Common difficulties:

Reading f(x) as f times x.
Losing signs: −x² with x = −3 is −(−3)² = −9, not 9.
Confusing f(x + 1) with f(x) + 1, and f(2x) with 2f(x).
Forgetting the second solution when a quadratic f(x) = k has two inputs.
Not expanding (x + 1)² correctly (it is x² + 2x + 1, not x² + 1).