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A. Quadratic Functions · Quadratic Functions

Function Notation: Evaluating and Interpreting f(x)

A2.2About 65 minutes
Learning goals
  • I can read and write function notation such as f(3) = 7.
  • I can evaluate linear and quadratic functions for numbers and simple expressions.
  • I can find the input that gives a known output.
  • I can explain what function notation means in a real-world context, with units.

Words to know

function notation
Writing f(x) for the output of the function f when the input is x. Read "f of x".
evaluate
Find the value of a function for a given input by substituting and simplifying.
f(a)
The output of f when the input is a. The point (a, f(a)) is on the graph of f.
y-intercept
The output when the input is 0, f(0).
zero of a function
An input x for which f(x) = 0; an x-intercept of its graph.

A shorter way to say it

"The height of the ball, in metres, 2 seconds after it was thrown." That is a lot of words. Mathematicians write h(2). The letter h names the function (height), the 2 in brackets is the input (time), and h(2) is the output (the height).

This is function notation. You will use it throughout your Grade 11 and Grade 12 math courses, and in physics, chemistry, economics and coding.

Reading function notation

For a function named f:

  • f(x) is read "f of x". It means the output of f when the input is x.
  • f(x) is not f times x. The brackets hold the input.
  • y = f(x). The graph of f is the set of points (x, f(x)).
  • f(3) = 7 says: when the input is 3, the output is 7. So the point (3, 7) is on the graph.

Functions can have any name: f, g, h, C for cost, P for profit, h for height.

f(3)= 7name of the functioninput: x = 3output: y = 7read "f of 3 equals 7" → the point (3, 7) is on the graph of f
Each part of f(3) = 7 has a job: f names the function, 3 is the input and 7 is the output. Together they describe one point on the graph.

Evaluating a function

To evaluate f(a), replace every x in the rule with a, then simplify.

Put brackets around the substituted value, especially when it is negative. For f(x) = x² − 3x, f(−2) = (−2)² − 3(−2) = 4 + 6 = 10. Without brackets, it is easy to write −2² = −4 by mistake.

Worked example

Example 1: Evaluating linear and quadratic functions

f(x) = 4x − 7 and g(x) = 2x² − 5x + 1. Find (a) f(3), (b) f(−1.5), (c) g(−2), (d) g(0) and (e) f(2) + g(3).

  1. (a) f(3) = 4(3) − 7 = 12 − 7 = 5.
  2. (b) f(−1.5) = 4(−1.5) − 7 = −6 − 7 = −13.
  3. (c) g(−2) = 2(−2)² − 5(−2) + 1 = 2(4) + 10 + 1 = 19.
  4. (d) g(0) = 2(0)² − 5(0) + 1 = 1. (For any function, f(0) is the y-intercept.)
  5. (e) g(3) = 2(9) − 15 + 1 = 4, so f(2) + g(3) = (8 − 7) + 4 = 1 + 4 = 5.

f(3) = 5, f(−1.5) = −13, g(−2) = 19, g(0) = 1, f(2) + g(3) = 5.

Worked example

Example 2: Substituting expressions

f(x) = x² − 3. Find (a) f(a), (b) f(2a) and (c) f(x + 2). Simplify.

  1. The rule is: square the input, then subtract 3. Whatever is in the brackets is the input.
  2. (a) f(a) = a² − 3.
  3. (b) f(2a) = (2a)² − 3 = 4a² − 3. Square the whole input: (2a)² = 4a², not 2a².
  4. (c) f(x + 2) = (x + 2)² − 3 = x² + 4x + 4 − 3 = x² + 4x + 1.

f(a) = a² − 3, f(2a) = 4a² − 3, f(x + 2) = x² + 4x + 1.

Working backwards: when does f(x) equal a value?

Sometimes you know the output and want the input. Set the rule equal to the output and solve.

  • A linear function usually gives one answer.
  • A quadratic function can give two answers, one answer or none, because a horizontal line can cross a parabola twice, once or not at all.
xy−3−2−1012345−5−4−3−2−101234567f(2) = −3(−2, 5)(4, 5)f(x) = 5f(x) = x² − 2x − 3
To find f(2), go up from x = 2 to the curve, then across to the y-axis: f(2) = −3. To solve f(x) = 5, draw the horizontal line y = 5. It crosses the parabola twice, so x = −2 or x = 4.
Worked example

Example 3: Solving f(x) = k

(a) f(x) = 3x + 4. Find x so that f(x) = 19.

(b) g(x) = x² − 2x − 3. Find x so that g(x) = 5.

  1. (a) 3x + 4 = 19, so 3x = 15 and x = 5. Check: f(5) = 15 + 4 = 19. ✓
  2. (b) x² − 2x − 3 = 5, so x² − 2x − 8 = 0.
  3. Factor (see the unit Quadratic Expressions and Equations): (x − 4)(x + 2) = 0, so x = 4 or x = −2.
  4. Check: g(4) = 16 − 8 − 3 = 5 ✓ and g(−2) = 4 + 4 − 3 = 5 ✓. The graph above shows both points.

(a) x = 5; (b) x = 4 or x = −2.

Worked example

Example 4: Function notation in context

A ball is tossed up from a balcony. Its height in metres after t seconds is h(t) = −4.9t² + 19.6t + 2.

(a) Find h(0), h(1) and h(2) and explain what each means.

(b) Explain in words what h(3) = 16.7 means.

  1. (a) h(0) = 0 + 0 + 2 = 2: the ball starts 2 m above the ground.
  2. h(1) = −4.9(1) + 19.6(1) + 2 = 16.7: after 1 s the ball is 16.7 m high.
  3. h(2) = −4.9(4) + 19.6(2) + 2 = −19.6 + 39.2 + 2 = 21.6: after 2 s it is 21.6 m high.
  4. (b) Check: h(3) = −4.9(9) + 58.8 + 2 = −44.1 + 60.8 = 16.7. ✓ After 3 s the ball is 16.7 m high, the same height as at 1 s. At 1 s it was going up; at 3 s it is coming down.

h(0) = 2, h(1) = 16.7, h(2) = 21.6. h(3) = 16.7 means the ball is 16.7 m high after 3 s.

Units tell the story

In a context, give every input and output its unit. "h(2) = 21.6" becomes "after 2 s, the height is 21.6 m". If C(n) is the cost in dollars of n items, then C(40) = 730 means "40 items cost $730".

Just for fun

Why is f(x) such a good listener?

You tell it one thing, and it always gives you exactly one answer back. No rambling.

Hands-on activity

The secret rule game

You need: a partner (a family member or friend), paper and a pencil. A calculator helps.

  1. Your partner secretly writes a rule, either linear (like f(x) = 3x − 1) or simple quadratic (like f(x) = x² + 2 or f(x) = 2x²).
  2. You call out inputs. Your partner answers with outputs, written in function notation: "f(4) = 11".
  3. Record the pairs in a table. After each answer, try to guess the rule.
  4. To check a guess, predict an output first, for example "f(10) should be 29", then ask.
  5. Swap roles.

Tips: asking for f(0) gives the constant term. Asking for f(1), f(2), f(3) lets you use first and second differences: constant first differences mean linear; constant second differences mean quadratic, and for steps of 1 the second difference is 2a.

What to look for: each guess should be written with function notation, and each guess should be tested with a prediction before it is accepted.

Your turn

1

f(x) = 3x − 5. Find f(4), f(−2) and f(0).

2

g(x) = x² + 2x − 1. Find g(3) and g(−3).

3

h(x) = −2x² + x. Find h(−1) and h(0.5).

4

f(x) = 4x + 1. Find x if f(x) = 21.

5

g(x) = x² − 9. Find x if g(x) = 16.

6

f(x) = x² + 1. Find f(a) and f(2a).

7

A cottage costs C(d) = 150d + 75 dollars to rent for d days (including a $75 cleaning fee). Find C(4) and explain its meaning.

More practice: warm-up

1

f(x) = x + 7. Find f(3).

2

f(x) = x². Find f(−4).

3

f(x) = 2x². Find f(3).

4

f(5) = 2. Which point is on the graph of f?

5

f(x) = 10 − x. Find f(10).

More practice: on your own

1

f(x) = −x² + 4x + 5. Find f(2) and f(−1). What does f(−1) = 0 tell you about the graph?

2

f(x) = 2x − 3 and g(x) = x². Find f(3) + g(−2) and 2f(1) − g(3).

3

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

4

A ball's height is h(t) = −5t² + 20t + 1 metres after t seconds. Find h(1) and h(3). Why are they equal?

5

f(x) = 3x + b and f(2) = 11. Find b, then find f(−1).

Challenge

1

f(x) = x² − 2x. Find f(5), then find every x with f(x) = f(5).

2

f(x) = ax² + 3 and f(2) = 23. Find a, then explain why f(−2) must also be 23.

3

A model rocket's height is h(t) = −4.9t² + 49t metres. Show that h(5 + d) = h(5 − d) for any number d. What does this tell you about the rocket's flight?

Check your understanding

Quiz

1.

f(x) = x² − 5. What is f(−3)?

2.

What does f(2) = 7 tell you?

3.

f(x) = 2x + 1. For which x is f(x) = 9?

4.

h(t) = −5t² + 30t gives a height in metres after t seconds. What is h(2)?

5.

g(x) = x² + x. What is g(a)?

6.

For any function f, the y-intercept of the graph of y = f(x) is…

For parents and teachers

Big idea: expectation A2.2 asks students to substitute into and evaluate linear and quadratic functions written in function notation, including functions from real-world situations.

How to help: ask your teen to read notation aloud ("f of 3") and to say what it means in words, with units, in a context: "C(12) = 300 means 12 tickets cost $300." Invent rules together (cost of a taxi ride, cost of a cottage rental) and evaluate them.

Common difficulties:

  • Reading f(x) as "f times x".
  • Sign errors when substituting negatives: insist on brackets, (−3)² = 9.
  • Substituting into only some of the x's in the rule.
  • Mixing up f(a) (find the output) with "f(x) = a" (find the input).
  • Forgetting the second answer when solving a quadratic f(x) = k.