A part of the graph of the function is shown below: Answer the following questions. Explain why ‘f’…
Mathematics20254 marksNumerical
A part of the graph of the function $f(x) = 2x^3 - 3x^2 - 12x + 8, x \in \mathbb{R}$ is shown below:
Answer the following questions.
(a)[2.0]
Explain why ‘f’ does not have an inverse.
(b)[2.0]
The domain of ‘f’ is now restricted to $a \le x \le b$ where $a < 0$ and $b > 0$. $a$ and $b$ are chosen so that f has an inverse and the interval $[a, b]$ is as large as possible. Find the domain and range of $f^{-1}$.
Answer
Answer (a)
Official answer key
It is clear from the graph that $f$ is not a one-to-one function. For example, $f(x) = 0$ corresponds to three different $x$ values (i.e., $f$ is many-to-one). As the graph shows, ‘$f$’ is not one-to-one. Therefore, it cannot have an inverse.
Answer (b)
Official answer key
Use derivative to find x-coordinates of turning points: $f'(x) = 6x^2 - 6x - 12 = 0 \Rightarrow x^2 - x - 2 = 0 \Rightarrow (x+1)(x-2) = 0 \Rightarrow x = -1, 2$
$\Rightarrow a = -1$, $b = 2$ $\Rightarrow f(a) = f(-1) = 15$, $f(b) = f(2) = -12$
So, ‘$f$’ has domain $-1 \leq x \leq 2$ and range $-12 \leq f(x) \leq 15$.
The domain of $f^{-1}$ is $-12 \leq x \leq 15$ and the range is $-1 \leq f^{-1}(x) \leq 2$.
