In fg(x), which function do I use first?

The one written closest to xxx. So fg(x)=f(g(x))fg(x)=f(g(x))fg(x)=f(g(x)) applies ggg first, then fff. Reading the letters right to left keeps the order straight.

Functions Worked Examples (Easy)

Q: What is the difference between an object and an image?

The object is the input you put into the function; the image is the output it produces. For f(x)=2x+3f(x)=2x+3f(x)=2x+3, the object 555 has image f(5)=13f(5)=13f(5)=13.

Q: What is the fastest way to check a composite value like gf(2)?

Do it two ways. Find gf(x)gf(x)gf(x) and substitute, and also work inside-out by computing f(2)f(2)f(2) first. If both routes give the same number, your answer is secure.

What these examples cover

These easy Functions examples build the four moves the whole chapter rests on: finding an image, finding an object, reading a range from a listed domain, and forming a simple composite. Each one uses small, clean numbers so you can follow every line without a calculator getting in the way.

The best way to use the set is to cover the solution, attempt the question in full on paper, and only then check line by line against our working. If your answer differs, find the exact step where the two solutions part company, that single line is usually where the real learning is.

Treat every substitution as a place to slow down, and these turn into quick, reliable marks.

Worked examples

Work through all four. Attempt each fully before you read the matching solution, and pay attention to how the same discipline, substitute carefully, keep brackets, simplify one step at a time, runs through every one.

Q1[3 marks]

Given the function $f(x)=2x+3$ , find (a) the image of $5$ , and (b) the object whose image is $21$ .

Show worked solution

(a) The image of $5$ is the output $f(5)$ . Substitute the object $x=5$ into the rule:

f(5)=2(5)+3=10+3=13

(b) An object whose image is $21$ is a value of $x$ with $f(x)=21$ . Set the rule equal to $21$ and solve for $x$ :

2x+3=21 \;\Rightarrow\; 2x=18 \;\Rightarrow\; x=9

Answer

The image of $5$ is $13$ , and the object is $9$ . Check part (b) by substituting back: $f(9)=2(9)+3=21$ , as required.

Q2[3 marks]

A function is defined by $g(x)=x^{2}-4x$ . Find (a) $g(3)$ and (b) $g(-2)$ .

Show worked solution

(a) Substitute $x=3$ into the rule and simplify:

g(3)=(3)^{2}-4(3)=9-12=-3

(b) Substitute $x=-2$ , and keep the negative object inside brackets before you square or multiply:

g(-2)=(-2)^{2}-4(-2)=4+8=12

Answer

$g(3)=-3$ and $g(-2)=12$ . The brackets around $-2$ are what keep both signs correct: $(-2)^{2}=4$ , not $-4$ , and $-4(-2)=+8$ .

Q3[3 marks]

The function $f$ is defined by $f:x \mapsto 5-2x$ with domain $\{-1, 0, 2, 4\}$ . Find the range.

Show worked solution

The range is the set of images produced by the four objects in the domain. Evaluate $f$ at each object in turn:

f(-1)=5-2(-1)=5+2=7

f(0)=5-2(0)=5

f(2)=5-2(2)=5-4=1

f(4)=5-2(4)=5-8=-3

Answer

The range is $\{7, 5, 1, -3\}$ . Because the domain here is a short list, the range is simply the list of outputs, one image for each object, with no interval to describe.

Q4[4 marks]

Two functions are defined by $f(x)=x+4$ and $g(x)=3x$ . Find (a) the composite function $fg(x)$ , and (b) the value of $gf(2)$ .

Show worked solution

(a) $fg(x)$ means $f(g(x))$ , so $g$ acts first. Replace the input of $f$ with the whole of $g(x)=3x$ :

fg(x)=f(3x)=3x+4

(b) $gf(x)$ means $g(f(x))$ , so now $f$ acts first. Work inside-out: find $f(2)$ , then apply $g$ to that result.

f(2)=2+4=6

gf(2)=g(6)=3(6)=18

Answer

$fg(x)=3x+4$ and $gf(2)=18$ . Check part (b) the other way: $gf(x)=g(x+4)=3(x+4)=3x+12$ , so $gf(2)=3(2)+12=18$ , which agrees.

Q5[2 marks]

A relation is defined by the ordered pairs $\{(1,3),(2,3),(3,5),(4,7)\}$ . State whether this relation is a function, and identify its type.

Show worked solution

A relation is a function only when every object has exactly one image. List each object with its image:

1\to3,\quad 2\to3,\quad 3\to5,\quad 4\to7

Every object above gives exactly one image, so this relation is a function. The objects $1$ and $2$ share the same image $3$ , which is the sign of a many-to-one relation.

Answer

This relation is a function, and it is many-to-one. A relation fails to be a function only when some object produces more than one image.

Q6[2 marks]

The function $h(x)=3x-k$ is such that $h(4)=7$ . Find the value of $k$ .

Show worked solution

Substitute $x=4$ into the rule, using the given image $7$ :

h(4)=3(4)-k=12-k=7

Solve this equation for $k$ :

12-k=7 \;\Rightarrow\; k=5

Answer

$k=5$ . Check: $h(4)=3(4)-5=12-5=7$ , which matches the given image.

Q7[3 marks]

The function $f$ is defined by $f(x)=2x-5$ . Find $f^{-1}(x)$ , and hence find $f^{-1}(3)$ .

Show worked solution

Write $y=f(x)$ , then rearrange to make $x$ the subject:

y=2x-5 \;\Rightarrow\; x=\dfrac{y+5}{2}

Replace $y$ with $x$ to state the inverse function, then substitute $x=3$ :

f^{-1}(x)=\dfrac{x+5}{2}, \qquad f^{-1}(3)=\dfrac{3+5}{2}=4

Answer

$f^{-1}(x)=\dfrac{x+5}{2}$ and $f^{-1}(3)=4$ . Check by working forward: $f(4)=2(4)-5=3$ , which confirms $3$ is the image of $4$ .

Q8[3 marks]

Given $g(x)=x+2$ and the composite function $fg(x)=3x+1$ , find $f(x)$ .

Show worked solution

Since $fg(x)=f(g(x))$ , replace $g(x)$ with $x+2$ inside $f$ , and set this equal to the given composite:

f(x+2)=3x+1

Let $u=x+2$ , so $x=u-2$ . Substitute to write the right side in terms of $u$ :

f(u)=3(u-2)+1=3u-5 \;\Rightarrow\; f(x)=3x-5

Answer

$f(x)=3x-5$ . Check: $fg(x)=f(x+2)=3(x+2)-5=3x+6-5=3x+1$ , which matches the given composite.

Notice how different these four questions look on the surface, yet how similar the discipline is underneath: read exactly what is asked, substitute carefully with brackets, and simplify one step at a time. That steadiness is what turns Functions from a source of small slips into a reliable opening block of marks.

Key method points

These four examples rehearse the everyday skills that open almost every Functions question in Add Math. Keep the following points in mind as you practise more.

To find an image, substitute the object into the rule; to find an object, set the rule equal to the image and solve.
Always wrap a negative object in brackets before you square or multiply, so every sign stays correct.
With a listed domain, the range is just the set of images, evaluate the rule at each object and collect the outputs.
In a composite $fg(x)$ , the function nearest $x$ acts first: $fg(x)=f(g(x))$ .
Check a composite value inside-out as an independent second route to the same answer.
Because marking is analytic, a clear substitution line can still earn method marks even if the final arithmetic slips.

How a teacher helps

When a student loses a mark on questions like these, it is almost always a small, fixable habit, a missing bracket around a negative object, or a composite read in the wrong order. In a one-to-one lesson our teacher watches the exact line where the slip happens and corrects it on the spot, before it settles into a routine.

Because our teachers are experienced, you work with someone who explains the why behind each step, not just the what. Lessons are taught in English, while SPM papers are set in both Malay and English, so we make sure the notation reads the same to you either way.

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Frequently asked questions

What is the difference between an object and an image?

The object is the input you put into the function; the image is the output it produces. For $f(x)=2x+3$ , the object $5$ has image $f(5)=13$ .

How do I find the range when the domain is given as a list?

Substitute each value in the domain into the rule, then collect the results. Those outputs, listed once each, form the range.

There is no interval to describe when the domain is a short set.

fg(x)

, which function do I use first?

The one written closest to $x$ . So $fg(x)=f(g(x))$ applies $g$ first, then $f$ .

Reading the letters right to left keeps the order straight.

What is the fastest way to check a composite value like

gf(2)

Do it two ways. Find $gf(x)$ and substitute, and also work inside-out by computing $f(2)$ first.

If both routes give the same number, your answer is secure.

Source:SRC-DSKP-EN

Written by the spmaddmath.com.my editorial team.· Last updated 5 September 2026

Functions, Worked Examples (easy)

What these examples cover

Worked examples

Key method points

How a teacher helps

Frequently asked questions

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