Category Archives: Functions

July 25, 2024 · 7:05 am

Function Composition and Domain and Range

My year 12 Specialist students have been working on function composition and the domain and range of the resulting composition. And they have been struggling a bit with why the composition doesn’t exist.

For example,

The functions $f$ and $g$ are defined by $f(x)=x^2-2$ and $g(x)=\sqrt{x}$

(a) Explain why $g(f(x))$ is not defined.

(b) By suitably restricting the domain of $f$ , obtain a function $f_1$ such that $g(f_1(x))$ is defined.

For the composite function to exist the range of the inner function (in this case $f(x)$ ) must be a subset of the domain of the outer function (in this case $g(x)$ ).

Start by finding the domain and range of each function.

f(x)=x^2-2

D_{f(x)}=\{x:x\in\mathbb{R}\}

We can see the range of $f(x)$ is not a subset of the domain of $g(x)$

i.e. $R_{f(x)}\nsubseteq D_{g(x)}$

We can restrict the range of $f(x)$ by restricting the domain.

$f(x)\geq0$

$x^2-2\geq0$

$x^2\geq2$

$x\leq -\sqrt{2}$ or $x\geq \sqrt{2}$

Therefore $f_1(x)=x^2-2, x\leq -\sqrt{2}$ or $x\geq \sqrt{2}$

and $g(f_1(x))=\sqrt{x^2-2}, x\leq -\sqrt{2}$ or $x\geq \sqrt{2}$

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Filed under Composition, Functions