Let ℎ(𝑥) = 2 − 4𝑥

Question

Let ℎ(𝑥) = 2 − 4𝑥

(a) Sketch the graph of ℎ(𝑥). On your graph, mark the points where

 ℎ(𝑥) = 0

 ℎ(0)

 ℎ(1)

b) Find the inverse of ℎ(𝑥).

(c) Show that the function you found in part (b) is the inverse of h by finding their composition.

Answer 1

the graph is a straight line with slope =-4 and y intercept 2 or the point (0,2)

the x intercept is when y=0 = 2-4x, 4x =2, x=2/4 = 1/2, the point (1/2,0)

plot the 2 intercepts and connect them with a straight line, extend it in both directions, putting an arrow at the end of the line, on both ends to indicate it goes forever

a) h(1/2) =0 that's the point (1/2,0) the x intercept

h(0) = 2 that's the point (0,2) the y intercept

h(1) = 2-4(1) = -2, the point (1,-2)

b) find the inverse by switching x and y and then solve for y

y=2-4x

x=2-4y

4y =2-x

y = (2-x)/4 = 1/2 -x/4

h^-1 = y = 1/2 - x/4

c) it's the inverse because if you start with a number x, such as 4 h(4) =2-4(4) =-14

and plug that into the inverse function h^-1(-14) = 1/2- (-14)/4 = 1/2 +7/2 = 8/2 = 4

you get back what you started with, 4

start with x, the composite function h^-1(f(x)) = x