The function f is defined by f(x) = e^{-x^4}, x ∈ R
Find f^{-1}(x) and state its domain. - AQA - A-Level Maths Pure - Question 4 - 2018 - Paper 1
Question 4
The function f is defined by f(x) = e^{-x^4}, x ∈ 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 ∈ R
Find f^{-1}(x) and state its domain. - AQA - A-Level Maths Pure - Question 4 - 2018 - Paper 1
Step 1
Find f^{-1}(x)
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Answer
To find the inverse of the function f, we start with the equation:
y=e−x4
Next, we will take the natural logarithm of both sides:
extln(y)=−x4
Rearranging gives:
x4=−extln(y)
Taking the fourth root results in:
oot{4}{- ext{ln}(y)} $$
Thus, we can express the inverse function as:
$$ f^{-1}(x) =
oot{4}{- ext{ln}(x)} $$
Step 2
State its domain
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Answer
The domain of the inverse function f−1(x) is determined by the range of the original function f(x). Since f(x)=e−x4 yields positive values for all real x, we conclude:
