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^{-... 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 true for all points of intersection if we remember that for a function and its inverse, they reflect across the line y=x. Thus, every point of intersection must satisfy f(x)=f−1(x), which occurs along the line.
Step 2
B. None of the points of intersection lie on the line $y = x$.
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Answer
This statement is false because, as previously discussed, points of intersection will lie on the line y=x.
Step 3
C. At no point of intersection are the tangents to the graphs parallel.
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Answer
This statement is likely true; for two graphs that are reflections of each other, their tangents would have slopes that are negative reciprocals of each other, which implies they cannot be parallel.
Step 4
D. At no point of intersection are the tangents to the graphs perpendicular.
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Answer
This statement is false. If the slopes of tangents are negative reciprocals at points of intersection, then they are perpendicular.
