Question

Identify the inverse g(x) of the given relation f(x). f(x) = {(8, 3), (4, 1), (0, –1), (–4, –3)}

Answer 1

Answer:

The inverse g(x) = 2x + 2

Step-by-step explanation:

* Lets explain the inverse of a function

- To find the inverse of any function we switch the x and y then

 we solve to find the new y

- The domain of the function is the values of x and the range of

 the function is the values of y

- The domain of the inverse function is the values of y and the

 range of the inverse function is the values of x

- Lets solve the problem

∵ f(x) = {(8 , 3) , (4 , 1) , (0 , -1) , (-4 , -3)}

- To find the inverse g(x) lets find f(x) from the order pairs

∵ x-coordinates are decreases by 4 and y-coordinates are

  decreases by 2

∴ The relation represents the linear function

- The form of the linear function is f(x) = mx + c , where m is the

  slope of the line and c is the y-intercept

∵ The slope of the line whose endpoints are (x1 , y1) and (x2 , y2)

  is m = (y2 - y1)/(x2 - x1)

- We can find the slope from any two order pairs

∵ (x1 , y1) = (8 , 3) and (x2 , y2) = (4 , 1)

∴ m = [1 - 3]/[4 - 8] = -2/-4 = 1/2

∵ f(x) = mx + c

∴ f(x) = 1/2 x + c

- The y-intercept means the line intersect the y-axis

  at point (0 , c)

∵ There is a point (0 , -1)

∴ c = -1

∴ f(x) = 1/2 x - 1

- To find the inverse of the function switch x and y and solve to

  find the new y

∵ y = 1/2 x - 1 ⇒ switch x and y

∴ x = 1/2 y - 1 ⇒ add 1 to both sides

∴ x + 1 = 1/2 y ⇒ Multiply both sides by 2

∴ 2(x + 1) = y

∴ y = 2x + 2

∵ g(x) is the inverse of f(x)

∵ The inverse of f(x) is 2x + 2

∴ g(x) = 2x + 2