Language of Functions 📘

Welcome, students, to a key lesson in Functions for IB Mathematics: Analysis and Approaches HL. This lesson is about the language of functions: how we describe them, read them, and use them in mathematics and real-world situations. Think of a function as a machine that takes in an input, follows a rule, and produces an output. In IB Mathematics, this idea appears everywhere: in graphs, formulas, tables, word problems, and modeling situations like population growth, revenue, or temperature change 📈

What you will learn

By the end of this lesson, students, you should be able to:

explain common function terms such as domain, range, input, output, and notation;
read and write function statements clearly using correct mathematical language;
interpret graphs, tables, and formulas as different representations of the same relationship;
understand how functions connect to polynomial, rational, exponential, and logarithmic models;
use the language of functions to describe transformations, inverses, and composites;
solve equations and inequalities involving functions in IB-style reasoning.

1. What is a function?

A function is a rule that assigns each input exactly one output. If the input is $x$, the output is often written as $f(x)$. The symbol $f(x)$ is read as “$f$ of $x$,” not “$f$ times $x$.” This notation is one of the most important parts of function language.

For example, if $f(x)=2x+3$, then:

$f(1)=2(1)+3=5$,
$f(4)=2(4)+3=11$.

This means the same rule is applied every time. The input changes, and the output changes according to the rule. A function is different from a relation that gives one input more than one output.

A helpful real-world example is a taxi fare. If the fare is modeled by $f(x)=4+1.5x$, where $x$ is the number of kilometers traveled, then $f(0)=4$ means there is a starting fee of $4$, and each extra kilometer adds $1.5$. The function language lets us describe the situation precisely 🚕

2. Domain, range, and notation

The domain of a function is the set of allowed inputs. The range is the set of possible outputs. These ideas matter because not every number can be used in every function.

For example, consider $g(x)=\frac{1}{x}$. Since division by $0$ is undefined, $x=0$ is not in the domain. So the domain is all real numbers except $0$.

For $h(x)=\sqrt{x-2}$, the expression inside the square root must be non-negative, so $x-2\ge 0$. This gives $x\ge 2$. Therefore the domain is $[2,\infty)$.

Function notation also helps when comparing values. If $f(a)=f(b)$, this means the function gives the same output for two different inputs. This can happen in many functions, especially non-linear ones. For instance, for $f(x)=x^2$, we have $f(2)=4$ and $f(-2)=4$.

IB exams often expect careful language. Instead of saying “$x$ can be anything,” it is better to state the domain clearly. Instead of saying “the graph goes up forever,” it is better to say the range is unbounded above.

3. Interpreting graphs, tables, and formulas

A function can be shown in several ways: as a formula, a table, a graph, or a description in words. Part of function language is translating between these forms.

If a graph shows a curve crossing the $x$-axis at $x=3$, that means $f(3)=0$. The $x$-intercepts are the values where the output is zero. The $y$-intercept occurs when $x=0$, so it is $f(0)$ if that value exists.

A table can show values such as:

$x=0$, $f(x)=2$,
$x=1$, $f(x)=5$,
$x=2$, $f(x)=10$.

From this table, students, you might notice the outputs are not increasing by a constant amount. That suggests the function may not be linear. Recognizing patterns is part of reading function language.

For example, if the outputs double each time, the model may be exponential. If the differences between outputs are constant, the model may be linear. If the second differences are constant, the model may be quadratic. These patterns help identify the type of function from data.

4. Polynomials, rational, exponential, and logarithmic functions

IB Mathematics includes several major families of functions, and the language of functions helps describe each one.

A polynomial function has the form

$$f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0,$$

where $a_n\ne 0$ and $n$ is a non-negative integer. Polynomial graphs are smooth and continuous. Their end behavior depends on the leading term $a_nx^n$.

A rational function is a quotient of polynomials, such as

$$r(x)=\frac{p(x)}{q(x)}.$$

The domain excludes values that make $q(x)=0$. Rational graphs may have vertical asymptotes, horizontal asymptotes, and holes.

An exponential function has the variable in the exponent, such as

$$f(x)=ab^x,$$

where $b>0$ and $b\ne 1$. Exponential functions are used for growth and decay, such as population growth, compound interest, and radioactive decay.

A logarithmic function is the inverse of an exponential function, such as

$$f(x)=\log_b(x).$$

Logarithms help solve equations where the variable is in the exponent. For example, solving $2^x=16$ leads to $x=4$, and more complex exponential equations may require logarithms.

Each family has a characteristic behavior. Being able to name and describe that behavior accurately is part of strong function language.

5. Transformations and inverse functions

Function language also includes transformations. If $f(x)$ is a basic graph, then transformations create a new function.

For example, $g(x)=f(x-3)+2$ means the graph of $f(x)$ shifts right by $3$ and up by $2$. The minus sign inside the function causes a horizontal shift in the opposite direction. This is an important IB idea because horizontal and vertical transformations behave differently.

Other common transformations include:

$g(x)=f(x)+k$, a vertical shift;
$g(x)=f(x-k)$, a horizontal shift;
$g(x)=af(x)$, a vertical stretch or compression;
$g(x)=f(ax)$, a horizontal stretch or compression;
$g(x)=-f(x)$, reflection in the $x$-axis;
$g(x)=f(-x)$, reflection in the $y$-axis.

Inverse functions reverse the effect of a function. If $f$ and $f^{-1}$ are inverses, then

$$f(f^{-1}(x))=x$$

and

$$f^{-1}(f(x))=x,$$

for values where both expressions are defined.

A simple example is $f(x)=3x-6$. To find the inverse, let $y=3x-6$, then solve for $x$:

$$x=\frac{y+6}{3}.$$

$$f^{-1}(x)=\frac{x+6}{3}.$$

Inverse functions are especially useful when a problem asks you to “undo” a process, such as converting Celsius to Fahrenheit and back again 🌡️

6. Composite functions and solving equations and inequalities

A composite function combines two functions. If $f$ and $g$ are functions, then

$$ (f\circ g)(x)=f(g(x)). $$

This means you apply $g$ first, then $f$.

For example, if $f(x)=x^2$ and $g(x)=x+1$, then

$$ (f\circ g)(x)=f(x+1)=(x+1)^2. $$

But

$$ (g\circ f)(x)=g(x^2)=x^2+1. $$

These are not the same, so order matters.

Composite functions appear naturally in modeling. Suppose the temperature in a city depends on time, and the electricity use depends on temperature. Then one function can feed into another.

Equations involving functions often ask when two outputs are equal. For example, solving $f(x)=g(x)$ means finding where two graphs intersect. If $f(x)=x^2$ and $g(x)=2x+3$, then solving

$$x^2=2x+3$$

gives the $x$-values where the graphs meet.

Inequalities involving functions ask where one function is greater than another. For example,

$$f(x)>g(x)$$

means the graph of $f$ lies above the graph of $g$.

This is useful in optimization, comparing profits, and finding when a quantity stays above or below a threshold.

Conclusion

The language of functions is the foundation of the whole Functions topic in IB Mathematics: Analysis and Approaches HL. students, when you understand terms like domain, range, function notation, inverse, and composite, you can read mathematics more accurately and solve problems more confidently. This language connects directly to polynomial, rational, exponential, and logarithmic models, and it supports graphing, transformations, and solving equations and inequalities. In IB work, clear function language is not extra decoration; it is part of the mathematics itself ✅

Study Notes

A function assigns each input exactly one output.
Write function values as $f(x)$, read as “$f$ of $x$.”
The domain is the set of allowed inputs; the range is the set of possible outputs.
Use correct notation for intercepts, such as $f(0)$ for the $y$-intercept.
Polynomial functions have the form $f(x)=a_nx^n+\dots+a_0$.
Rational functions are quotients of polynomials, like $r(x)=\frac{p(x)}{q(x)}$.
Exponential functions have the variable in the exponent, such as $f(x)=ab^x$.
Logarithmic functions are inverses of exponential functions, such as $f(x)=\log_b(x)$.
Transformations include shifts, stretches, and reflections.
Inverse functions satisfy $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$.
Composite functions are written as $(f\circ g)(x)=f(g(x))$.
Solving $f(x)=g(x)$ finds intersections; solving $f(x)>g(x)$ compares outputs.
Clear language and accurate notation are essential in IB Mathematics: Analysis and Approaches HL.