Understand · explore · practise
Understand functions, domain and range, one-to-one and many-to-one mappings. Explore restricted quadratics and practise ranges of roots, reciprocals, exponentials and logarithms.
Before you startFunction notation, quadratic graphs, reciprocals and basic exponential/logarithmic graphs
01 / What counts as a function?
A function assigns one output to each input in its stated domain. Different inputs are allowed to share an output. What is not allowed is one input with two outputs, or an input in the domain with no output.
Function: every input in the domain → exactly one output
For example, f(x) = x² is a function on all real numbers: f(−3) = f(3) = 9 causes no problem. In contrast, the relation y² = x does not specify a function of x on positive inputs unless you choose one branch: at x = 4 it permits both y = 2 and y = −2.
The notation √x already chooses the nonnegative square root. Thus y = √x is a function with real domain x ≥ 0.
02 / Domain versus range
The domain is the set of allowed inputs. The range is the set of outputs actually reached from that domain. State ranges in terms of y or f(x), not by relabelling the input restrictions.
f(x) = x² − 1
Domain: {−2, −1, 0, 1, 2}
Evaluate each allowed input.
Outputs in that order:
3, 0, −1, 0, 3
Outputs may repeat.
Range: {−1, 0, 3}
List each achieved value once. There are five inputs and three distinct outputs.
03 / Finding allowed inputs
When no smaller domain is specified, find the real inputs for which the formula is defined. A denominator cannot be zero, an even root needs a nonnegative argument, and a real logarithm needs a strictly positive argument.
2/(x + 3): x ≠ −3
All other real inputs are allowed.
√(5 − x): x ≤ 5
Zero is allowed inside a square root.
ln(x − 2): x > 2
Zero is not allowed inside a logarithm.
A stated domain can be narrower than these natural restrictions. Keep the stated restriction when simplifying a formula; cancelling a denominator factor does not automatically restore its excluded input.
04 / Quadratic ranges
Completing the square gives f(x) = x² − 4x + 7 = (x − 2)² + 3. On all real inputs its minimum is 3, attained at x = 2, so the range is [3, ∞).
On the restricted domain −1 ≤ x ≤ 4, the vertex is still included. The endpoints give f(−1) = 12 and f(4) = 7. The range is therefore [3, 12]. Checking only endpoints would miss the minimum.
On 3 < x ≤ 5, the function is increasing and the vertex is excluded. The output approaches 4 as x approaches 3 from above, but never equals 4. At x = 5 it reaches 12. The range is (4, 12].
Green: allowed part of f(x) = (x − 1)² + 2. Blue D: input interval. Gold R: output interval. Filled endpoints are included.
Domain: −2 ≤ x ≤ 4. Range: 2 ≤ y ≤ 11.
This restriction is many-to-one: inputs on opposite sides of the vertex can share an output.
Pause, replay or seek freely. The notes explain the same idea and stay in view.
05 / Reciprocals, roots and exponentials
Use monotonicity and boundary behaviour. If the graph approaches an asymptote forever, its asymptote value is not automatically part of the range.
h(x) = 3/(x − 1), 1 < x ≤ 4
At x = 4 the output is 1. As x approaches 1 from above it increases without bound. Range: [1, ∞).
g(x) = √(5 − x), x ≤ 5
At x = 5 the output is 0; smaller inputs give arbitrarily large outputs. Range: [0, ∞).
p(x) = 2ˣ − 3, x real
Since 2ˣ > 0 and takes every positive value, the range is (−3, ∞).
q(x) = ln(x − 2), x > 2
Its argument takes every positive value, so the logarithm takes every real value. Range: all real y.
06 / One-to-one or many-to-one
A one-to-one function never gives the same output to two different allowed inputs. A many-to-one function does. Both satisfy the definition of a function.
On a graph, the vertical-line test checks whether each input has exactly one output, provided the graph covers the stated domain. Once that is established, the horizontal-line test checks whether an output is shared by different inputs.
The quadratic f(x) = (x − 1)² + 2 is many-to-one on all real inputs. Restrict it to x ≥ 1, and it becomes one-to-one because it increases throughout that branch. Restricting to x ≤ 1 also works, using the decreasing branch.
If the domain is x ≥ a, the smallest a that makes this quadratic one-to-one is 1. Any a < 1 includes some pair 1 − t and 1 + t with t > 0, giving the same output. With a = 1, no such pair remains.
One-to-one behaviour is the condition needed to reverse a function on its range. A later lesson develops inverse functions.
07 / Your turn
Distinguish open endpoints from closed ones, and explain whether a proposed output can actually occur.
Does f(x) = |x| define a function on all real numbers? Is it one-to-one?
Compare inputs 2 and −2, then remember the definition.
It is a function: each real input has exactly one modulus. It is many-to-one, because f(2) = f(−2) = 2. Two inputs sharing one output do not violate the function definition.
Find the range of f(x) = 2x² + 1 for domain {−2, 0, 1, 2}. Is f one-to-one on this domain?
Evaluate the four inputs and remove duplicate outputs.
The outputs are 9, 1, 3 and 9. The range is {1, 3, 9}. The function is many-to-one because −2 and 2 both produce 9.
Find the real domain of (a) 1/(2x − 5), (b) √(x + 4) and (c) ln(7 − x).
Use nonzero, nonnegative and strictly positive conditions respectively.
(a) x ≠ 5/2. (b) x ≥ −4. (c) x < 7. The logarithm endpoint is excluded, while the square-root endpoint is allowed.
Find the range of f(x) = (x + 1)² − 2 for −3 ≤ x ≤ 2.
The vertex is inside this interval.
The minimum is −2 at x = −1. Endpoint values are f(−3) = 2 and f(2) = 7. By continuity the function reaches every value between minimum and maximum, so the range is [−2, 7].
Find the range of f(x) = (x + 1)² − 2 for −1 < x ≤ 2.
The minimum of the unrestricted graph is now approached but not attained.
The function increases on this interval. Outputs approach −2 as x approaches −1, but x = −1 is excluded. The maximum 7 is attained at x = 2. Range: (−2, 7].
Find the range of g(x) = 2/(x + 1) for x ≥ 1.
Check x = 1 and the limit as x grows.
At x = 1 the output is 1. All outputs are positive and decrease towards zero without reaching it. Range: (0, 1].
Find the range of p(x) = 2ˣ − 3 for x ≤ 0.
Here 0 < 2ˣ ≤ 1.
Subtracting 3 gives −3 < p(x) ≤ −2. Range: (−3, −2]. The upper endpoint occurs at x = 0; the lower is never attained.
Find the least a for which h(x) = x² + 6x + 11 is one-to-one on x ≥ a. State its range for that least a.
Complete the square.
h(x) = (x + 3)² + 2 has vertex at x = −3. The least a is −3; the range on x ≥ −3 is [2, ∞). Any smaller a includes inputs on both sides of the vertex sharing an output.
Does x² + y² = 4 define y as a function of x on −2 ≤ x ≤ 2? Give a restriction that would make it a function.
At x = 0, inspect the possible y-values.
No: x = 0 has both y = 2 and y = −2. Restricting to y ≥ 0 gives the upper semicircle y = √(4 − x²), a function on −2 ≤ x ≤ 2 with range [0, 2].
Find the range of q(x) = ln(x − 2) for 2 < x ≤ 3.
The argument ranges over (0, 1].
The logarithm is increasing, reaches 0 at argument 1, and is unbounded below as the argument approaches zero. Range: (−∞, 0].
08 / Recap
Section 1 of 8 · What counts as a function?