Year 11 Maths - Specialist Graphing techniques

Reciprocal Functions

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Questions

Question 1

58612

Sketch the graphs of $f(x)=1-x^2$ and $g(x)=\dfrac{1}{f(x)}$ on the same set of axes.

Answer

Refer to worked solution

Worked Solution

$$\begin{aligned}
&\begin{aligned}
f(x)&=1-x^{2} & \\
\text {Let }\ f(x)&=0\ (1-x)(1+x)=0 \\
\therefore x&=\pm 1
\end{aligned}\\
&\text{Asymptotes for}\ g(x)=\frac{1}{1-x^{2}}\\
&\text{when}\ x=0 \quad f(0)=g(0)=1\\
&\text { for }\ x>1\ f(x) \rightarrow-\infty, g(x) \rightarrow 0 \\
&\quad x<1 \ f(x) \rightarrow-\infty, g(x) \rightarrow 0\\
&\text{In the domain}\ -1<x<1\\
&\text { for }
x \rightarrow 1,\ g(x) \rightarrow+\infty \\
&\quad x \rightarrow-1\ g(x) \rightarrow-\infty
\end{aligned}$$

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