Answer:
[tex]f^{-1}(6) = 2[/tex]
Step-by-step explanation:
Given
[tex]f(2) = 6[/tex]
[tex]f(3) = 7[/tex]
Required
[tex]f^{-1}(6)[/tex]
First, we need to determine the slope of the function using;
[tex]m = \frac{y_1 - y_2}{x_1 - x_2}[/tex]
From the given parameters;
In [tex]f(2) = 6[/tex]
x = 2; y =6 --- Take this as x1 and y1
In [tex]f(3) = 7[/tex]
x = 3; y = 7 --- Take this as x2 and y2
[tex]m = \frac{y_1 - y_2}{x_1 - x_2}[/tex] becomes
[tex]m = \frac{6 - 7}{2 - 3}[/tex]
[tex]m = \frac{- 1}{ - 1}[/tex]
[tex]m = 1[/tex]
Next, we determine the equation of the function using
[tex]y - y_1 = m(x - x_1)[/tex]
Substitute the values of x1,y1 and m
[tex]y - 6 = 1(x - 2)[/tex]
Open bracket
[tex]y - 6 = x - 2[/tex]
Add 6 to both sides
[tex]y - 6 + 6 = x -2 +6[/tex]
[tex]y = x + 4[/tex]
Next is to determine the inverse function by swapping the positions of x and y
[tex]x = y + 4[/tex]
Make y the subject of formula;
[tex]y = x - 4[/tex]
Replace y with [tex]f^{-1}(x)[/tex]
[tex]f^{-1}(x) = x - 4[/tex]
Now, we can solve for [tex]f^{-1}(6)[/tex]
Substitute 6 for x
[tex]f^{-1}(6) = 6 - 4[/tex]
[tex]f^{-1}(6) = 2[/tex]
