A given function $f(x)$ has an inverse $f^{-1}(x)$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2022 - Paper 1
Question 9
A given function $f(x)$ has an inverse $f^{-1}(x)$.
The derivatives of $f(x)$ and $f^{-1}(x)$ exist for all real numbers $x$.
The graphs $y=f(x)$ and $y=f^{-1}... show full transcript
Worked Solution & Example Answer:A given function $f(x)$ has an inverse $f^{-1}(x)$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2022 - Paper 1
Step 1
A. All points of intersection lie on the line $y=x$.
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Answer
This statement is not true because for y=f(x) and y=f−1(x) to intersect, they must satisfy f(x)=x, but not all intersections must fall exclusively on this line.
Step 2
B. None of the points of intersection lie on the line $y=x$.
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This statement is also not true, as it is possible for the graphs to intersect at points that do lie on the line y=x, specifically at points where f(x)=x.
Step 3
C. At no point of intersection are the tangents to the graphs parallel.
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This is incorrect because the tangents could be parallel at some points of intersection if the derivatives of both functions are equal at those points.
Step 4
D. At no point of intersection are the tangents to the graphs perpendicular.
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This statement is true because the derivative of the inverse function satisfies the relationship (f^{-1})'(x) = rac{1}{f'(f^{-1}(x))}. For the tangents to be perpendicular, the product of the slopes would need to equal −1, which is not the case for those intersections.
