TMUA Functions and Graph Transformations

One-to-one and many-to-one

A function is one-to-one if different inputs always give different outputs. $x^2$ is many-to-one on all real numbers but one-to-one if the domain is restricted to $x\ge0$.

Composite functions

$f(g(x))$ means apply $g$ first, then $f$. In general $f(g(x))\ne g(f(x))$.

Transformations

• $f(x)+a$ moves the graph up by $a$.
• $f(x+a)$ moves it left by $a$.
• $af(x)$ stretches vertically by $a$; $f(ax)$ stretches horizontally by $\tfrac1a$.

Sketching and counting roots

To count the real roots of a polynomial, sketch it: find where it meets the axes and where it turns. A cubic can have one, two or three real roots, while a quadratic has at most two.

Know the shapes of $x^2$, $x^3$, $\tfrac1x$, $\sqrt x$, $|x|$ and $\sin x$. For a transformation, track two or three key points, such as a vertex and an intercept, instead of redrawing the whole curve.

Worked examples

Common mistakes

• Shifting the wrong way: $f(x+a)$ moves left, not right.
• Applying a vertical stretch to the $x$-coordinate of a point.
• Composing in the wrong order.
• Choosing a domain that includes both sides of the vertex and calling the function one-to-one.

Test yourself on this topic with our free practice questions.