Composite and inverse function explorer

This tool composes two functions in both orders, finds their inverses where an inverse exists and checks the result by going forward and back. It helps students who know the notation but lose marks on the order of the functions or on the domain.

How do I use it?

1. Choose the type of function f: linear (px + q), square (px² + q) or reciprocal (p/x + q). Enter p and q.
2. Choose its domain: all real numbers, x ≥ 0 only, or x ≤ 0 only. Do the same for g.
3. Enter an input value x. The starting example is f(x) = 2x + 1 and g(x) = x² with x = 3.
4. Press Work it out and read the composition steps, then the inverse section.
5. Change the domain of g to x ≥ 0 and press the button again to see an inverse appear.

How does the tool work?

Every value is computed directly from the rule you entered. The tool never guesses a formula.

For f(g(x)), it finds g(x) first, then puts that value into f. For g(f(x)), it finds f(x) first, then puts that value into g. If any step is outside a domain, uses a square root of a negative number or divides by zero, the tool says the value is undefined.

An inverse is offered only when the function is one-to-one on the domain you chose. The rule for each family is:

• Linear: f⁻¹(y) = (y − q) ÷ p.
• Square on x ≥ 0: f⁻¹(y) = √((y − q) ÷ p). On x ≤ 0 the square root takes a minus sign.
• Reciprocal: f⁻¹(y) = p ÷ (y − q), with y ≠ q.

The domain of the inverse is the range of the original function, and the tool prints it.

What does a worked example look like?

Take f(x) = 2x + 1 and g(x) = x², with x = 3. For f(g(3)), g(3) = 9 and then f(9) = 19.

For g(f(3)), f(3) = 7 and then g(7) = 49. The two answers, 19 and 49, are different, so the order matters.

For the inverse, f⁻¹(y) = (y − 1) ÷ 2. The check gives f(3) = 7 and f⁻¹(7) = 3, which returns to the start. For g on all real numbers the tool reports no single inverse, because g(3) = g(−3) = 9.

What can the tool not tell you?

It covers three families of function only, and it is not a general symbolic solver. It does not produce a simplified formula for f(g(x)), and it does not cover exponential, logarithmic or trigonometric functions.

It also cannot decide the domain for an exam question. Always read the domain stated in the question and write it in your answer, because the composite and the inverse both depend on it.

How does it connect to the lessons?

Begin with finding composite functions in the correct order. Then move to finding inverse functions with domain restrictions, and try the functions practice.

The reciprocal family links to solving inverse variation problems and to finding the inverse of a two by two matrix. The square family connects to the quadratic functions practice.

If you want a teacher to go through the gaps, see Additional Mathematics tuition. A one-hour trial class (from RM50) is a real taught lesson.