Function Worksheets: Evaluating, Inverses & Transformations

What it is

A function is a rule that assigns exactly one output to each input. f(x) is read 'f of x' and names that output for the input x. The inverse function f⁻¹(x) undoes f: swap x and y, then solve for y — for a linear function this always gives back another function, but for a quadratic it gives a ± relation with two outputs for most inputs, a function only once the original's domain is restricted. A transformation y = af(k(x − d)) + c stretches, reflects, and shifts the graph of a parent function like x, x², √x, or 1/x.

Why it matters

Functions are the language of higher math and science; reading f(x) notation and evaluating it fluently is the gateway to graphing, modelling, and calculus. Inverses and transformations build the graphical intuition that everything from trigonometry to exponential models leans on.

How to do it

1. To evaluate: replace every x in the rule with the given input value, and simplify.
2. To find the inverse: swap x and y, then solve the resulting equation for y.
3. A quadratic's inverse: swap and solve the same way, but expect a ± square root — that's a relation, not a function, unless the domain is restricted.
4. To describe a transformation: check the sign of a and k for reflections, |a| and |k| for stretches, then d and c for shifts.

Examples by level

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

• Treating f(x) as f times x instead of 'f of x'.
• Substituting the input in some terms but not all.
• Sign errors when the input is negative (forgetting (−3)² = 9).
• Solving for the inverse but forgetting to swap x and y first.
• Mixing up which parameter controls a horizontal vs. a vertical change (k affects x, a affects y, even though both can look like a 'stretch').
• Assuming a quadratic's inverse is automatically a function — swap-and-solve gives a ± relation (two y-values for most x), a function only once the original's domain is restricted.

Tips

• f(x) is not f × x; it's the name of the rule applied to x.
• Wrap negative inputs in parentheses to keep signs and exponents right.
• In y = af(k(x − d)) + c, the letter k affects x (horizontal) and a affects the output (vertical), even though both can look like a 'stretch'.
• A quadratic's 'inverse' from swap-and-solve is usually a ± relation — write both branches rather than picking just one.

Key vocabulary

function

a rule giving each input exactly one output

input

the value substituted for x

inverse function

the function that undoes f, found by swapping x and y and solving for y

relation

a set of input-output pairs; unlike a function, a relation can pair one input with more than one output — a quadratic's unrestricted inverse is a relation, not a function

parent function

the simplest form of a function family, e.g. f(x) = x² before any transformation

Frequently asked questions

What does f(x) mean?

It's function notation ('f of x') naming the output of the rule f for the input x. f(4) means the output when x is 4.

How do you evaluate a function?

Replace every x in the rule with the given number and simplify.

How do you find the inverse of a function?

Swap x and y, then solve the resulting equation for y. For f(x) = mx + b, the inverse is f⁻¹(x) = (x − b) / m.

Is the inverse of a quadratic function also a function?

No — swapping x and y in y = (x − h)² + k and solving gives y = h ± √(x − k), a ± relation with two outputs for most inputs, not a function. It only becomes a function if the original quadratic's domain is restricted first (e.g., to x ≥ h).

What do a, k, d, and c control in y = af(k(x − d)) + c?

a stretches/reflects vertically, k stretches/reflects horizontally, d shifts horizontally, and c shifts vertically.