Solving function equations and inequalities

IB Mathematics: Analysis and Approaches HL· 2.5 Functions· 5 min read

1. Solving Function Equations Algebraically★★★☆☆⏱ 15 min

📘 Definition

Function equation

An equation that involves one or more functions, requiring either the form of an unknown function or input values that satisfy the relation

Example:

$f (x + 1) = 2 x + 3$ or $f (2 x) = f (x)^{2}$

Most IB exam questions ask you to find input values that satisfy an equation involving a known function. Common strategies include substitution of composite inputs, equating coefficients for unknown functions, and using function properties like odd/even symmetry.

📐 Worked Example

Given $f (x) = x^{2} - 4 x$ , solve $f (x + 1) = 5$

1
Substitute $(x + 1)$ into the function definition and expand:
2
$f(x+1) = (x+1)^2 - 4(x+1) = x^2 - 2x - 3$
3
Set equal to 5 and rearrange to standard quadratic form:
4
$x^2 - 2x - 3 = 5 \implies x^2 - 2x - 8 = 0$
5
Factor and solve for $x$ :
6
$(x-4)(x+2) = 0 \implies x = 4, \; x = -2$
7
$f (x)$ is a polynomial with domain all real numbers, so both solutions are valid.

✓ Quick check

Which of the following is a function equation?
- $3 x + 5 = 0$
- $f (x) = 3 x + 5$
- $f (x + 2) = 4 x - 1$
- $y = 2 x^{2}$
Reveal answer
2 —
A is a linear equation in $x$ , B is a function definition, C is a function equation that asks for either $x$ or the form of $f$ .

2. Domain Restrictions and Extraneous Solutions★★★★☆⏱ 15 min

When solving function equations, the original function has a fixed domain. Any solution that lies outside this domain is extraneous and must be rejected. Common sources of restrictions include square roots, denominators, and logarithms.

📐 Worked Example

Solve $2 x + 5 = x + 1$ for real $x$

1
First identify domain restrictions: the expression under the square root must be non-negative, and the left side (a square root) is non-negative, so the right side must also be non-negative:
2
$2x + 5 \geq 0 \implies x \geq -\frac{5}{2}, \quad x + 1 \geq 0 \implies x \geq -1$
3
Square both sides to eliminate the square root:
4
$(\sqrt{2x + 5})^2 = (x+1)^2 \implies 2x + 5 = x^2 + 2x + 1$
5
Simplify and solve:
6
$x^2 - 4 = 0 \implies x = 2, \; x = -2$
7
Check against the domain: $x = - 2 < - 1$ , so it is extraneous. Only $x = 2$ is valid. Verify in the original equation to confirm.

3. Solving Function Inequalities★★★★☆⏱ 20 min

📘 Definition

Function inequality

An inequality that requires finding all input values that satisfy a relation between two function expressions

Example:

$\frac{1}{x} > x$

Two methods are common: algebraic interval testing and graphical intersection. The key to avoiding mistakes is to never multiply both sides by a variable term that can be negative, as this flips the inequality sign unpredictably.

📐 Worked Example

Solve $\frac{1}{x} > x$ for real $x \neq = 0$

1
Rearrange all terms to the left side:
2
$\frac{1}{x} - x > 0 \implies \frac{1 - x^2}{x} > 0 \implies \frac{(1-x)(1+x)}{x} > 0$
3
Identify critical points (where numerator/denominator is zero): $x = - 1, 0, 1$ . These split the number line into 4 intervals.
4
Test the sign of the expression in each interval: 1. $x < - 1$ : negative → no solution; 2. $- 1 < x < 0$ : positive → solution; 3. $0 < x < 1$ : negative → no solution; 4. $x > 1$ : positive → solution.
5
The final solution set is $- 1 < x < 0$ or $x > 1$ .

4. Common Pitfalls

Wrong move:

Multiplying both sides of a function inequality by a variable expression without flipping the inequality sign for negative values

Why:

Variable expressions can be positive or negative, and the inequality sign flips when multiplying by a negative

Correct move:

Rearrange all terms to one side to get an expression greater/less than zero, then test intervals around critical points

Wrong move:

Forgetting to reject extraneous solutions after solving a radical or rational function equation

Why:

Algebraic steps like squaring both sides introduce solutions that do not satisfy the original equation

Correct move:

Always check all solutions against the original equation and its domain before writing your final answer

Wrong move:

Including points where the function is undefined in the solution set of an inequality

Why:

Points of discontinuity or vertical asymptotes are not part of the domain, so they cannot satisfy the inequality

Correct move:

Exclude all undefined points from your solution, use open intervals around asymptotes

Wrong move:

Incorrect substitution of composite inputs in function equations

Why:

Mixing up substitution order leads to incorrect starting equations and wrong solutions

Correct move:

Substitute the inner input step by step, expand and simplify carefully before solving

5. Quick Reference Cheatsheet

Strategy	Use Case	Key Exam Tip
Algebraic substitution	Solve $f (g (x)) = k$	Check domain after solving
Interval testing	All function inequalities	Rearrange to 0 on one side first
Graphical intersection	Find $f (x) > g (x)$	Mark roots/asymptotes before reading intervals
Equating coefficients	Find unknown function $f$	Match all powers, including constants

6. Frequently Asked

Do I always need to check for extraneous solutions?

Yes, any time you square both sides, multiply by an expression that can equal zero, or work with functions with restricted domains, you must check all solutions in the original equation to avoid losing marks.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Solve composite function equation
2022 · 2
Solve rational function inequality
2023 · 1
Find solutions to radical function equation

What's Next

This sub-topic builds on core function properties you will use throughout the entire IB AA HL course, from polynomial algebra to differential calculus. Mastery of solving function equations and identifying extraneous solutions prepares you for more advanced topics like composite functions, inverse functions, and finding roots of higher-degree polynomials in upcoming units. You will also regularly apply these skills when solving optimization problems in calculus, where you need to determine where functions change sign to identify critical points, maxima, and minima. The habit of checking domain restrictions you develop here will help you avoid common mistakes in nearly every other topic you study for the exam.