The function $f$ is defined by $f(x) = e^{-x^4}, x \in \mathbb{R}$
Find $f^{-1}(x)$ and state its domain. - AQA - A-Level Maths Mechanics - Question 4 - 2018 - Paper 1
Question 4
The function $f$ is defined by $f(x) = e^{-x^4}, x \in \mathbb{R}$
Find $f^{-1}(x)$ and state its domain.
Worked Solution & Example Answer:The function $f$ is defined by $f(x) = e^{-x^4}, x \in \mathbb{R}$
Find $f^{-1}(x)$ and state its domain. - AQA - A-Level Maths Mechanics - Question 4 - 2018 - Paper 1
Step 1
Take logs of an equation. Must be correct use of logs.
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Answer
Starting from the equation:
y=e−x4
We take the natural logarithm of both sides:
ln(y)=−x4
Step 2
Obtain correct inverse function in any correct form.
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Answer
Rearranging gives:
−ln(y)=x4
From here, we take the fourth root:
x=4−ln(y)
Thus, the inverse function can be expressed as:
f−1(x)=4+ln(x), for x>0
Step 3
Deduce correct domain.
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Answer
The domain of the inverse function f−1(x) is that x must be positive:
