Question
The graph of y = f (x) for 0 ≤ x ≤ 10 is shown in the following diagram.
The graph intercepts the axes at (10, 0) and (0, 5).
(a) Write down the value of
(i) \( f(4) \)
(ii) \( f \circ f(4) \)
(iii) \( f^{-1}(3) \)
(b) On the axes above, sketch the graph of \( y = f^{-1}(x) \). Show clearly where the graph intercepts the axes.
▶️Answer/Explanation
Detailed solution
(a) (i) \( f(4) \)
– To find \( f(4) \), locate \( x = 4 \) on the x-axis and trace vertically to the curve.
– From the graph, when \( x = 4 \), the y-value appears to be 1.
– So, \( f(4) = 1 \).
(ii) \( f \circ f(4) \) means \( f(f(4)) \).
– First, we found \( f(4) = 1 \).
– Now, find \( f(1) \) by locating \( x = 1 \) on the x-axis and tracing vertically to the curve.
– From the graph, when \( x = 1 \), the y-value appears to be 3.
– So, \( f \circ f(4) = f(1) = 3 \).
(iii) \( f^{-1}(3) \) is the x-value where \( f(x) = 3 \).
– Locate \( y = 3 \) on the y-axis and trace horizontally to the curve.
– From the graph, when \( y = 3 \), the x-value appears to be 1.
– So, \( f^{-1}(3) = 1 \).
(b) Sketching the Graph of \( y = f^{-1}(x) \)
– The inverse function \( y = f^{-1}(x) \) is obtained by swapping the x and y coordinates of the original function \( y = f(x) \).
– The original graph intercepts the axes at \( (0, 5) \) and \( (10, 0) \).
– For the inverse:
– When \( x = 0 \), \( y = 5 \), so the inverse intercepts the y-axis at \( (0, 5) \).
– When \( x = 10 \), \( y = 0 \), so the inverse intercepts the x-axis at \( (10, 0) \).
– Since \( f(x) \) is a decreasing function, \( f^{-1}(x) \) will also be decreasing.
– The shape of \( y = f^{-1}(x) \) will be a reflection of \( y = f(x) \) over the line \( y = x \). Given the original curve’s shape, the inverse will be a similar decreasing curve starting at \( (0, 5) \) and ending at \( (10, 0) \). The graph of \( f(x) \) and \( f^{-1}(x) \) is as follows
………………….Markscheme…………………..
(a)
(i) \( f(4) = 1 \)
(ii) \( f \circ f(4) = 3 \)
(iii) \( f^{-1}(3) = 1 \)
(b).
A concave up curve with:
– \( y \)-intercept at \( (0, 10) \)
– \( x \)-intercept at \( (5, 0) \)
the curve passes through (2,2) OR through \( (1, 4) \) and \( (3, 1) \).