Introduction to Functions
This topic covers the fundamental definition of a function as a mapping from inputs to outputs, where each input has only one output. It focuses on distinguishing between one-to-one and many-to-one functions and understanding the properties of the square root and modulus functions.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- A function is a rule that assigns exactly one output value to each unique input value.
- A 'many-to-one' function can produce the same output from different inputs (e.g., f(x) = x2).
- A 'one-to-one' function produces a unique output for every unique input (e.g., f(x) = x3).
- The expression f(x) = √(x) conventionally refers to the POSITIVE square root only. Its domain is x ≥ 0.
- The modulus function f(x) = |x| returns the non-negative value of x. The graph of y = |f(x)| is obtained by reflecting any part of the graph of y = f(x) that is below the x-axis, into the positive y-region.
- Graphically, a curve represents a function if any vertical line crosses it at most once (the 'vertical line test').
Diagram
Formulae
|x| = x if x ≥ 0, and |x| = -x if x < 0 This is the piecewise definition of the modulus function, useful for solving equations or inequalities involving absolute values by splitting them into cases.
Definitions
- Function
- A mathematical relationship between a set of inputs and a set of outputs, where each input is related to exactly one output.
- Many-to-one
- A property of a function where at least two different inputs map to the same output. For example, in f(x) = x2, the inputs x=2 and x=-2 both map to the output y=4.
- One-to-one
- A property of a function where every distinct input produces a distinct output. No output value is ever repeated.
Worked example
The function g(x) is defined as g(x) = 5 - |x + 2|. Which of the following statements is true about g(x)? A) It is a one-to-one function. B) The equation g(x) = 6 has two distinct solutions. C) The maximum value of g(x) is 5. D) The graph of y = g(x) does not cross the y-axis.
- 1
1.
First, visualise the graph of y = |x + 2|.
This is a 'V' shape with its vertex (minimum point) at (-2, 0).
- 2
2.
Next, consider y = -|x + 2|.
This reflects the 'V' shape in the x-axis, turning it into an upside-down 'V' (an 'A' shape) with its vertex (maximum point) at (-2, 0).
- 3
3.
Finally, consider y = 5 - |x + 2|.
This translates the entire graph from step 2 upwards by 5 units.
The vertex is now at (-2, 5).
- 4
4.
The graph is an 'A' shape with a maximum point at y=5.
Therefore, the function is many-to-one, as any horizontal line below y=5 will cross the graph twice.
This rules out A.
- 5
5.
The maximum value of the function is the y-coordinate of the vertex, which is 5.
Therefore, the equation g(x) = 6 can never be satisfied, as g(x) ≤ 5 for all x.
This rules out B.
- 6
6.
From step 5, the maximum value is indeed 5.
This makes statement C a candidate.
- 7
7.
To check the y-intercept, we set x = 0.
g(0) = 5 - |0 + 2| = 5 - 2 = 3The graph crosses the y-axis at (0, 3).
This rules out D.
- 8
8.
Therefore, the only true statement is C.
Answer: C
Common mistakes
- ×Mistaking the equation y2 = x for the function y = √(x). The former is a 'sideways parabola' and not a function, while the latter is only the top half.
- ×Confusing the definition of a function (one y for each x) with the definition of a one-to-one function (one x for each y). A parabola y=x2 is a function, but it is many-to-one.
- ×Incorrectly applying transformations to the modulus function, for example by shifting `|x|` before applying the negative sign in a function like `5 - |x+2|`.
- ×Forgetting that `√(x)` is undefined for negative real numbers, which can lead to errors when determining the domain of a function.
No-calculator tips
- ✓Use the 'Vertical Line Test' to quickly determine if a graph is a function. Imagine sweeping a vertical line across the plot; if it ever touches the curve in more than one place, it's not a function.
- ✓Use the 'Horizontal Line Test' on a function's graph to classify it. If any horizontal line can cross the curve more than once, the function is many-to-one.
- ✓Sketching graphs is the fastest way to solve problems about modulus functions. Remember `y=|f(x)|` involves reflecting the negative parts of `f(x)` upwards.