Domain, range and ways to define a function
A function is a rule that gives exactly one output for each input. We write y = f(x).
- Domain: all x values we are allowed to put in.
- Range: all y values that come out.
A function can be given by a formula (f(x) = 2x + 1), a table, a graph or words ("double it and add one"). On a graph, the domain is the shadow of the curve on the x-axis, and the range is its shadow on the y-axis.
Example: for f(x) = x² − 2x − 3, the domain is all real numbers and the range is y ≥ −4.
Zeros and intervals of constant sign
A zero of f is an x where f(x) = 0. On the graph, it is where the curve meets the x-axis.
Between zeros, the graph stays on one side of the axis. So f keeps the same sign there. These are intervals of constant sign.
For f(x) = x² − 2x − 3 = (x + 1)(x − 3): zeros at −1 and 3. f(x) > 0 for x < −1 and for x > 3. f(x) < 0 for −1 < x < 3.
Increasing and decreasing (monotonicity)
f is increasing on an interval if a bigger x always gives a bigger f(x): x₁ < x₂ ⇒ f(x₁) < f(x₂). The graph goes up as you move right.
f is decreasing if x₁ < x₂ ⇒ f(x₁) > f(x₂). The graph goes down as you move right.
A function that is only increasing (or only decreasing) on an interval is called monotonic there. f(x) = 2x + 1 is increasing everywhere. f(x) = x² − 2x − 3 is decreasing for x ≤ 1 and increasing for x ≥ 1.
Tip: always state intervals using x values, not y values.
Maximum and minimum values
The maximum value of f is the largest f(x) it reaches. The minimum value is the smallest. On the graph they are the highest and lowest points.
Where f changes from increasing to decreasing, there is a local maximum (top of a hill). Where it changes from decreasing to increasing, there is a local minimum (bottom of a valley).
For x² − 2x − 3 = (x − 1)² − 4, the smallest value is −4 at x = 1, because a square is never negative. There is no maximum.
For x³ − 3x: local maximum 2 at x = −1, local minimum −2 at x = 1.
Even and odd functions (parity)
First check that the domain is symmetric (if x is in it, so is −x). Then:
- Even: f(−x) = f(x). The graph is a mirror image in the y-axis. Examples: x², x² − 4, |x|.
- Odd: f(−x) = −f(x). The graph looks the same after a half-turn (180°) about the origin. Examples: x, x³, x³ − 3x.
- Most functions are neither, like x² − 2x − 3.
Test: put −x in place of x and simplify. If you get f(x) back, it is even; if you get −f(x), it is odd.
Try it: predict, then check
In the 3D, pick f(x) = x³ − 3x. Before you slide x, guess: where is it increasing? Where is it negative? Then slide x from −4 to 4 and watch the readout. Make a table of x and f(x) on paper for x = −2, −1, 0, 1, 2 and check it is odd: f(−1) = 2 and f(1) = −2.
Key formulas and definitions
- Zero: f(x) = 0
- Increasing: x₁ < x₂ ⇒ f(x₁) < f(x₂)
- Decreasing: x₁ < x₂ ⇒ f(x₁) > f(x₂)
- Even: f(−x) = f(x)
- Odd: f(−x) = −f(x)
- Vertex form: a(x − h)² + k has min k (a > 0) or max k (a < 0) at x = h
Worked examples
1. Find the zeros of f(x) = 3x − 6.
Set 3x − 6 = 0 ⇒ 3x = 6 ⇒ x = 2. The only zero is x = 2.
2. Find the zeros and sign intervals of f(x) = x² − 9.
x² − 9 = (x − 3)(x + 3) = 0 ⇒ x = −3 or 3. Test x = 0: f(0) = −9 < 0. So f < 0 on (−3, 3), and f > 0 for x < −3 and x > 3.
3. Is f(x) = −2x + 5 increasing or decreasing?
Take x₁ = 0, x₂ = 1: f(0) = 5, f(1) = 3. Bigger x gave smaller f. The slope −2 is negative, so f is decreasing everywhere.
4. Find the minimum value of f(x) = x² − 6x + 11.
Complete the square: x² − 6x + 9 + 2 = (x − 3)² + 2. A square is ≥ 0, so f ≥ 2. Minimum value 2 at x = 3. Decreasing for x ≤ 3, increasing for x ≥ 3.
5. Is f(x) = x⁴ + x² even, odd or neither?
f(−x) = (−x)⁴ + (−x)² = x⁴ + x² = f(x). So it is even.
6. Is f(x) = x³ + x + 1 even, odd or neither?
f(−x) = −x³ − x + 1. This is not f(x), and −f(x) = −x³ − x − 1 is different too (the +1). So neither.
Common mistakes
- Writing increasing intervals with y values. Intervals of monotonicity are always about x.
- Saying the minimum is at x = 1 when asked for the minimum value. The value is f(1) = −4.
- Calling a function even just because it has x². Test f(−x): x² − 2x is not even.
- Forgetting to check the domain is symmetric before testing parity. f(x) = √x is neither.