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 $y = f^{-1}(x)$ have at least one point of intersection. Which statement is true for all points of intersection of these graphs? A. All points of intersection lie on the line $y = x$. B. None of the points of intersection lie on the line $y = x$. C. At no point of intersection are the tangents to the graphs parallel. D. At no point of intersection are the tangents to the graphs perpendicular.

SSCE HSC Mathematics Extension 1 Solving differential equations using separation of variables: A given function $f(x)$ has an i

A given function $f(x)$ has an inverse $f^{-1}(x)$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2022 - Paper 1

Question 9

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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 the points of intersection of the function and its inverse. For a function and its inverse to intersect, they must lie on the line $y = x$ .

Step 2

B. None of the points of intersection lie on the line $y = x$.

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Answer

This statement is incorrect since the intersection points must satisfy the equation of 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 can be false depending on the behavior of the function. If the derivative of the function at the intersection point is equal to the derivative of the inverse function at the same point, the tangents can 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 true. For two curves to be perpendicular at an intersection point, the product of their slopes (derivatives) must equal $-1$ . However, since the slopes of a function and its inverse are reciprocals, they cannot be perpendicular at their points of intersection.

A given function $f(x)$ has an inverse $f^{-1}(x)$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2022 - Paper 1

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

A. All points of intersection lie on the line $y = x$.

B. None of the points of intersection lie on the line $y = x$.

C. At no point of intersection are the tangents to the graphs parallel.

D. At no point of intersection are the tangents to the graphs perpendicular.

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