Question

Based on the graph given for f (x) construct the graph of f¹ (x). for f (x) find E, F and the formula of f (x) when it is known that it is a linear function.​

Answer 1

Answer:

To find an inverse function, reflect a graph of a function across the line y=x (and find the resulting equation)

To reflect a linear function in the line y=x, find points on f(x) and then swap their x and y coordinates.

Points on f(x):  (0, -4)  (2, 0)  (5, 6)

Points reflected in line y=x:  (-4, 0)  (0, 2)  (6, 5)

Plot points (-4, 0)  (0, 2)  (6, 5) and connect to form a straight line - this is the inverse of the function:  $f^{-1}(x)$

To determine the equation of f(x):

Choose 2 points on f(x):  (5, 6) and (0, -4)

Calculate the slope (gradient) by using:

$m=\dfrac{\triangle y}{\triangle x}=\dfrac{y_2-y_1}{x_2-x_1}=\frac{6--4}{5-0}=2$

Using the slope-intercept form:  y = mx + b

(where m is the slope and b is in the y-intercept)

From inspection of the graph, we can see the line crosses the y-axis at -4,

⇒ f(x) = 2x - 4

As the line is actually a line segment (with endpoints (0, -4) and (5, 6), then

f(x) = 2x - 4,   0 ≤ x ≤ 5

To determine the equation of $f^{-1}(x)$:

Rewrite f(x)  as  $y = 2x - 4$

Swap the x and y:  $x = 2y - 4$

Rearrange to make y the subject: $y= \dfrac{1}{2}(x + 4)$

Replace y with $f^{-1}(x)$

So the equation of the inverse is:   $f^{-1}(x)= \dfrac{1}{2}(x + 4)$

As the original function is a segment, then

$f^{-1}(x)= \dfrac{1}{2}(x + 4), \ \ \ -4\leq x\leq 6$

(shown in blue on the attached diagram)

** I can't see any points labelled E and F on the original function. If they are the endpoints of the line segment of f(x), then they are (0, -4) and (5, 6) **