What Is a Function?

A function is a rule, machine or mapping that takes an input and produces one output. If the input is \(x\), the output is commonly written as \(f(x)\), read as “\(f\) of \(x\).” The key condition is simple but strict: every input in the domain must have exactly one output in the range.

The notation:

\[ f: x \mapsto f(x) \]

means that the function \(f\) maps the input \(x\) to the output \(f(x)\). For example:

\[ f(x)=2x+3 \]

If \(x=5\), then:

\[ f(5)=2(5)+3=13 \]

In words, the function takes \(5\), doubles it, adds \(3\), and returns \(13\).

Function Mapping Diagram

Vertical Line Test

A graph represents a function if every vertical line intersects the graph at no more than one point. The reason is that a vertical line fixes one \(x\)-value. If that line touches the graph twice, one input has two outputs, so the relation is not a function.

Quick Classification of Function Types

Functions can be classified in several ways. One method groups them by formula: linear, quadratic, polynomial, rational, radical, exponential, logarithmic and trigonometric. Another method groups them by behaviour: increasing, decreasing, even, odd, periodic, one-to-one, many-to-one, bounded, unbounded, continuous or discontinuous. A third method groups them by construction: piecewise functions, composite functions, inverse functions and transformed functions.

1. Constant Functions

A constant function has the same output for every input. Its general form is:

\[ f(x)=c \]

The graph is a horizontal line. If \(f(x)=4\), every \(x\)-value gives \(4\). The domain is usually all real numbers, written as \(\mathbb{R}\), and the range is the single value \(\{4\}\).

Constant functions are useful because they model quantities that do not change with the input. In exam questions, they often appear when students compare gradient, rate of change, horizontal lines and special cases of linear functions. A constant function has gradient \(0\).

\[ f(x)=4,\quad \text{domain } \mathbb{R},\quad \text{range } \{4\} \]

2. Identity Functions

The identity function returns the input unchanged:

\[ f(x)=x \]

It is a straight line through the origin with slope \(1\). It is called the identity function because applying it does not change the value. For example, \(f(7)=7\) and \(f(-3)=-3\).

This function becomes important when studying inverse functions because a function and its inverse “undo” each other:

\[ f(f^{-1}(x))=x \quad \text{and} \quad f^{-1}(f(x))=x \]

3. Linear Functions

A linear function has the form:

\[ f(x)=mx+c \]

Here, \(m\) is the slope or gradient, and \(c\) is the \(y\)-intercept. The slope measures how much the output changes when the input increases by \(1\). If \(m>0\), the function increases. If \(m<0\), the function decreases. If \(m=0\), the function becomes constant.

The slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is:

\[ m=\frac{y_2-y_1}{x_2-x_1} \]

Linear functions model fixed-rate situations such as simple cost calculations, distance at constant speed, currency conversion, temperature conversion and straight-line depreciation. In exams, students are often asked to find the equation of a line, interpret the slope, identify the intercept, or solve two linear functions simultaneously.

4. Quadratic Functions

A quadratic function has degree \(2\). The standard form is:

\[ f(x)=ax^2+bx+c,\quad a\ne0 \]

The graph of a quadratic function is a parabola. If \(a>0\), the parabola opens upward and has a minimum point. If \(a<0\), it opens downward and has a maximum point. Quadratic functions appear in projectile motion, area problems, optimization, revenue models, braking distance, and many algebra questions.

The vertex form is:

\[ f(x)=a(x-h)^2+k \]

The vertex is \((h,k)\). The axis of symmetry is:

\[ x=h \]

In standard form, the \(x\)-coordinate of the vertex is:

\[ x=-\frac{b}{2a} \]

The quadratic formula solves \(ax^2+bx+c=0\):

\[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]

The discriminant tells the number of real roots:

\[ \Delta=b^2-4ac \]

• If \(\Delta>0\), there are two distinct real roots.
• If \(\Delta=0\), there is one repeated real root.
• If \(\Delta<0\), there are no real roots.

5. Polynomial Functions

A polynomial function is built from powers of \(x\) with non-negative integer exponents:

\[ f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 \]

The degree \(n\) is the highest power with a non-zero coefficient. Polynomial graphs are smooth and continuous. They do not have breaks, holes or vertical asymptotes. The leading term \(a_nx^n\) controls the end behaviour.

For even degree polynomials, both ends of the graph move in the same direction. For odd degree polynomials, the ends move in opposite directions. The sign of the leading coefficient determines whether the right-hand end rises or falls.

6. Cubic Functions

A cubic function is a polynomial of degree \(3\):

\[ f(x)=ax^3+bx^2+cx+d,\quad a\ne0 \]

Cubic graphs often have an S-shape, although the exact shape depends on coefficients. A simple parent cubic is:

\[ f(x)=x^3 \]

The parent cubic is odd because:

\[ f(-x)=(-x)^3=-x^3=-f(x) \]

Cubic functions are important in graph sketching, transformations, modelling volume, and studying rates of change. In exams, students may need to factor cubics, identify intercepts, sketch end behaviour, or interpret turning points.

7. Absolute Value Functions

The absolute value function is:

\[ f(x)=|x| \]

It measures distance from zero, so it is never negative. The graph is V-shaped with a sharp corner at the origin. A transformed absolute value function can be written as:

\[ f(x)=a|x-h|+k \]

The vertex is \((h,k)\). If \(a>0\), the V opens upward. If \(a<0\), it opens downward. Absolute value functions are used in distance problems, error measurement, tolerances and piecewise definitions.

Absolute value can also be written as a piecewise function:

\[ |x|= \begin{cases} x, & x\ge0\\ -x, & x<0 \end{cases} \]

8. Rational Functions

A rational function is a quotient of two polynomials:

\[ f(x)=\frac{p(x)}{q(x)},\quad q(x)\ne0 \]

The denominator cannot be zero. This restriction creates excluded values in the domain. Rational functions may have vertical asymptotes, horizontal asymptotes, oblique asymptotes or holes. A simple parent rational function is:

\[ f(x)=\frac{1}{x} \]

Its domain is \(x\ne0\), and its range is \(y\ne0\). The graph has two branches, one in Quadrant I and one in Quadrant III.

For a rational function, vertical asymptotes often occur where the denominator equals zero after simplification:

\[ q(x)=0 \]

Rational functions model rates, inverse variation, average cost, pressure-volume relationships and situations where division by a changing quantity matters.

9. Radical and Square Root Functions

A radical function contains a root expression. The most common parent function is:

\[ f(x)=\sqrt{x} \]

For real-valued square root functions, the expression under the square root must be non-negative:

\[ x\ge0 \]

A transformed square root function has the form:

\[ f(x)=a\sqrt{x-h}+k \]

Its starting point is usually \((h,k)\), and the domain is \(x\ge h\) if the expression is \(x-h\). Radical functions appear in geometry, physics, distance formulas, inverse quadratic relationships and algebraic equation solving.

10. Exponential Functions

An exponential function has the variable in the exponent:

\[ f(x)=ab^x,\quad b>0,\quad b\ne1 \]

If \(b>1\), the function grows exponentially. If \(0

\[ f(x)=e^x \]

Exponential functions model compound interest, population growth, radioactive decay, cooling, bacteria growth, inflation and algorithmic scaling.

Compound growth can be written as:

\[ A=P(1+r)^t \]

Continuous growth can be written as:

\[ A=Pe^{rt} \]

The horizontal asymptote of \(f(x)=b^x\) is:

\[ y=0 \]

11. Logarithmic Functions

A logarithmic function is the inverse of an exponential function. The basic form is:

\[ f(x)=\log_b(x),\quad b>0,\quad b\ne1,\quad x>0 \]

The logarithm answers the question: “What exponent is needed?” For example:

\[ \log_2(8)=3 \quad \text{because} \quad 2^3=8 \]

Logarithmic functions have domain \(x>0\). The parent graph has a vertical asymptote at:

\[ x=0 \]

Useful logarithm rules include:

\[ \log_b(MN)=\log_b(M)+\log_b(N) \]

\[ \log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N) \]

\[ \log_b(M^k)=k\log_b(M) \]

Logarithms appear in pH, sound intensity, earthquake magnitude, data scaling, finance and exponential equation solving.

12. Trigonometric Functions

Trigonometric functions connect angles with ratios and cycles. The three core functions are:

\[ \sin x,\quad \cos x,\quad \tan x \]

Sine and cosine are periodic waves with period \(2\pi\):

\[ \sin(x+2\pi)=\sin x \]

\[ \cos(x+2\pi)=\cos x \]

Tangent has period \(\pi\):

\[ \tan(x+\pi)=\tan x \]

A transformed sine function is often written as:

\[ f(x)=a\sin(b(x-h))+k \]

Here, \(|a|\) is the amplitude, \(h\) is the phase shift, \(k\) is the vertical shift, and the period is:

\[ \text{Period}=\frac{2\pi}{|b|} \]

Trigonometric functions model waves, sound, light, seasons, tides, circular motion, alternating current and repeated patterns.

13. Piecewise Functions

A piecewise function uses different formulas for different parts of the domain:

\[ f(x)= \begin{cases} x^2, & x<0\\ x+1, & x\ge0 \end{cases} \]

Piecewise functions are useful when one rule cannot describe the whole situation. Examples include tax brackets, shipping costs, parking fees, mobile data plans, step functions, absolute value functions and real-world restrictions.

When working with a piecewise function, first decide which interval the input belongs to. Then use the matching formula. For example, if:

\[ f(x)= \begin{cases} x^2, & x<0\\ x+1, & x\ge0 \end{cases} \]

then \(f(-3)=(-3)^2=9\), but \(f(3)=3+1=4\).

14. Composite Functions

A composite function is created by putting one function inside another. The notation:

\[ (f\circ g)(x)=f(g(x)) \]

means “apply \(g\) first, then apply \(f\).” If:

\[ f(x)=x^2 \quad \text{and} \quad g(x)=x+3 \]

then:

\[ (f\circ g)(x)=f(x+3)=(x+3)^2 \]

But:

\[ (g\circ f)(x)=g(x^2)=x^2+3 \]

In general:

\[ f(g(x))\ne g(f(x)) \]

Composite functions are important in transformations, modelling multi-step processes, calculus chain rule preparation and inverse functions.

15. Inverse Functions

An inverse function reverses the effect of the original function. If \(f\) sends \(x\) to \(y\), then \(f^{-1}\) sends \(y\) back to \(x\). The inverse relationship is:

\[ f^{-1}(f(x))=x \]

To find an inverse function:

1. Write \(f(x)\) as \(y\).
2. Swap \(x\) and \(y\).
3. Rearrange to make \(y\) the subject.
4. Write the result as \(f^{-1}(x)\).

Example:

\[ f(x)=2x+3 \]

\[ y=2x+3 \]

Swap \(x\) and \(y\):

\[ x=2y+3 \]

Rearrange:

\[ x-3=2y \]

\[ y=\frac{x-3}{2} \]

Therefore:

\[ f^{-1}(x)=\frac{x-3}{2} \]

Even, Odd, One-to-One and Many-to-One Functions

Functions can also be classified by symmetry and input-output behaviour.

Parent Functions and Transformations

A parent function is the simplest version of a function family. Transformations create new graphs from parent graphs. The most useful transformation formula is:

\[ g(x)=a f\big(b(x-h)\big)+k \]

Each parameter has a clear effect:

Domain and Range Rules

The domain is the set of allowed input values. The range is the set of possible output values. Domain and range are essential because many function types have restrictions. A student who ignores restrictions may get an algebraically neat answer that is mathematically invalid.

Course and Exam Alignment

Types of functions is not a standalone exam. It is a core algebra and precalculus topic that appears across many curricula, including school algebra, GCSE, IGCSE, A-Level, IB Mathematics, AP Precalculus, SAT Math, ACT Math and first-year university preparation. Because each board has its own syllabus, exam timing and grade boundaries, this page does not pretend that one official global score table exists.

Instead, use the following RevisionTown mastery score table to measure readiness for function questions. This is a learning rubric, not an official grade boundary.

7-Day Study Timetable for Types of Functions

Since there is no single official exam date for this topic, use this timetable as a practical preparation plan. It can be adapted for a one-week revision sprint, a classroom lesson sequence, or a self-study plan before a test.

Mastery Checklist

Tick what you can do. The progress tool gives a quick readiness score for this topic.

Practice Questions

1. State whether \(y^2=x\) is a function of \(x\). Explain using the vertical line test.
2. Find \(f(4)\) when \(f(x)=3x-7\).
3. Find the domain of \(f(x)=\frac{1}{x-6}\).
4. Find the domain of \(g(x)=\sqrt{x+5}\).
5. Identify the vertex of \(f(x)=2(x-3)^2-4\).
6. Find the inverse of \(f(x)=5x-2\).
7. If \(f(x)=x^2\) and \(g(x)=x+1\), find \(f(g(x))\).
8. State the period of \(y=3\sin(2x)\).
9. Write \(|x-2|\) as a piecewise function.
10. Explain why \(f(x)=\ln(x-4)\) has domain \(x>4\).

Answers

1. It is not a function of \(x\), because most positive \(x\)-values give two \(y\)-values: \(y=\pm\sqrt{x}\).
2. \(f(4)=3(4)-7=5\).
3. \(x\ne6\).
4. \(x\ge-5\).
5. The vertex is \((3,-4)\).
6. \(f^{-1}(x)=\frac{x+2}{5}\).
7. \(f(g(x))=(x+1)^2\).
8. \(\text{Period}=\frac{2\pi}{2}=\pi\).
9. \[ |x-2|= \begin{cases} x-2, & x\ge2\\ 2-x, & x<2 \end{cases} \]
10. The input of a logarithm must be positive, so \(x-4>0\), which gives \(x>4\).

Common Mistakes to Avoid

Frequently Asked Questions

Conclusion

Understanding types of functions is one of the most important foundations in mathematics. Once you know how to recognize a function family, read its formula, predict its graph, find its domain and range, and apply transformations, many algebra and precalculus topics become easier. Linear functions teach constant change. Quadratic functions introduce turning points. Polynomial functions develop end behaviour. Rational functions introduce restrictions and asymptotes. Radical functions strengthen domain thinking. Exponential and logarithmic functions explain growth, decay and inverse relationships. Trigonometric functions model cycles. Piecewise, composite and inverse functions teach structure and deeper reasoning.

The best way to revise is not to memorize isolated formulas only. Study the connection between formula, graph, domain, range and behaviour. When those four parts work together, function questions become far more manageable.