ℎ(𝑥)=3𝑥+4, hello i need help solving the inverse!

Question

ℎ(𝑥)=3𝑥+4

a.)Find the inverse function ℎ^−1(𝑥).

b.) Graph both h(x) and h^-1(x)

Answer 1

h(x) = 3x +4

y = 3x + 4

switch x and y and solve for the new y

x = 3y +4

3y =x -4

y = (x-4)/3 = x/3 - 4/3

h^-1(x) = x/3 - 4/3 or (x-4)/3

h(x) is an upward sloping straight line with slope =3 and y intercept = 4

h^-1(x) is a downward slopeing straight line with slope =1/3 and y intercept = -4/3

when (x,y) is a point on the h(x) line, (y,x) is a point on the h^-1(x) line. the inverse function just flips the x and y coordinates.

draw a diagonal line with slope =1 or 45 degrees with the x axis. h(x) is the mirror reflection of h^-1(x) if you put a mirror on the diagonal line

the inverse reverses the arithmetic operations.

h(x) = 3x +4

h^-1(x) = (x-4)/3

h(x) multiplies x by 3 then adds 4

h(x) subtracts 4 then divides by 3

addition is the inverse of subtraction

divisiion is the inverse of multiplication

h(h^-1(x)) = x

3((x-4)/3 + 4 = x-4+4 = x

or

h^-1(h(x)) = x

((3x+4)-4)/3 = 3x/3 = x

you start with x, do the 1st function, then the inverse function and you get back to where you started: x

Answer 2

h(x) = 3x + 4

x = 3 h^-1(x) + 4

x - 4 = 3 h^-1(x)

(x - 4) / 3 = h^-1(x)

Graph using techniques from Algebra 1.