Year 12 Functions and relations: Sum, difference and product of functions

Combined functions are built by adding, subtracting, or multiplying two existing functions. The main Year 12 skill is not just finding the new formula. It is understanding how the combination changes the graph. A sum might lift or lower parts of the shape. A product might create new zeros because one factor becomes zero. These questions reward you for thinking both algebraically and graphically at the same time.

Build the new function first, then ask what important graphical features changed: zeros, symmetry, turning behaviour, and sign.

Main combinations

$(f + g) (x) = f (x) + g (x)$
$(f - g) (x) = f (x) - g (x)$
$(f g) (x) = f (x) g (x)$

Worked example 1

Problem: Let

f (x) = x^{2}

and

g (x) = x - 1

. Find

(f + g) (x)

Add the two functions: $(f + g) (x) = x^{2} + (x - 1)$ .
Simplify: $(f + g) (x) = x^{2} + x - 1$ .

Answer:

(f + g) (x) = x^{2} + x - 1

Worked example 2

Problem: Let

f (x) = x - 2

and

g (x) = x + 3

. Find

(f g) (x)

and its x-intercepts.

Multiply the functions: $(f g) (x) = (x - 2) (x + 3)$ .
The product is zero when either factor is zero.
So the x-intercepts occur at $x = 2$ and $x = - 3$ .
If expanded, the product is $x^{2} + x - 6$ .

Answer:

(f g) (x) = (x - 2) (x + 3)

, with x-intercepts at

2

and

- 3

Why products are different

Products deserve extra attention because multiplying functions changes zeros and sign in a distinctive way. If one factor is zero, the whole product is zero. If both factors are positive, the product is positive. If one is positive and the other negative, the product is negative. Reading a product graph often begins with reading the sign of each factor first.

Worked example 3

Problem: Let

f (x) = x + 1

and

g (x) = x^{2}

. Find

(f - g) (x)

and determine its y-intercept.

Subtract the second rule from the first: $(f - g) (x) = (x + 1) - x^{2}$ .
Write the result clearly as $(f - g) (x) = - x^{2} + x + 1$ .
The y-intercept occurs when $x = 0$ .
Substitute to get $(f - g) (0) = 1$ .

Answer:

(f - g) (x) = - x^{2} + x + 1

and the y-intercept is

(0, 1)

Common traps

Treating $f (g (x))$ like a product instead of a composition.
Adding or multiplying formulas correctly but not simplifying them cleanly.
Forgetting that a product inherits zeros from either factor.

Revision focus

Combined-function questions are a good reminder that algebraic form and graph behaviour are linked. A product written in factored form tells you more about intercepts. A sum or difference in expanded form may tell you more about degree and symmetry. Choosing the more useful form is part of the skill, not an optional extra.

In revision, try sketching rough sign charts for products of simple functions. That habit makes it much easier to predict where the graph sits above or below the axis before doing detailed plotting.

It is equally useful to compare what happens under addition and subtraction. A difference can reverse local behaviour because subtracting a large positive value is the same as adding a negative one. This is why a clean algebraic simplification step is worth doing before you try to reason from the graph.

Practice links

Sketching sum, difference and product of two functions