Question

Suppose a function f has an inverse. If f(2)=6 and f(3)=7, find: f−1(6)

Answer 1

Answer:

$f^{-1}(6) = 2$

Step-by-step explanation:

Given

$f(2) = 6$

$f(3) = 7$

Required

$f^{-1}(6)$

First, we need to determine the slope of the function using;

$m = \frac{y_1 - y_2}{x_1 - x_2}$

From the given parameters;

In $f(2) = 6$

x = 2; y =6 --- Take this as x1 and y1

In $f(3) = 7$

x = 3; y = 7 --- Take this as x2 and y2

$m = \frac{y_1 - y_2}{x_1 - x_2}$ becomes

$m = \frac{6 - 7}{2 - 3}$

$m = \frac{- 1}{ - 1}$

$m = 1$

Next, we determine the equation of the function using

$y - y_1 = m(x - x_1)$

Substitute the values of x1,y1 and m

$y - 6 = 1(x - 2)$

Open bracket

$y - 6 = x - 2$

Add 6 to both sides

$y - 6 + 6 = x -2 +6$

$y = x + 4$

Next is to determine the inverse function by swapping the positions of x and y

$x = y + 4$

Make y the subject of formula;

$y = x - 4$

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

$f^{-1}(x) = x - 4$

Now, we can solve for $f^{-1}(6)$

Substitute 6 for x

$f^{-1}(6) = 6 - 4$

$f^{-1}(6) = 2$

Answer 2

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