Answer:
The inverse g(x) = 2x + 2
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