Question 298195: How do I find the inverse of the following function?
g(x) = -(1/5)x+1 Found 2 solutions by palanisamy, Theo:Answer by palanisamy(496) (Show Source):
You can put this solution on YOUR website! Put g(x) = y
Then, y = -(1/5)x + 1
y-1 = -(1/5)x
-5(y-1) = x
So, the required inverse function is -5(x-1)
You solve for x and then you invert x and y as follows:
Let y = g(x).
Equation becomes:
y = -(1/5)*x + 1
Solve for x as follows:
Subtract 1 from both sides of the equation to get:
y - 1 = -(1/5)*x
Multiply both sides of the equation by 5 to get:
5 * (y-1) = -x
Multiply both sides of the equation by -1 to get:
-5 * (y-1) = x
Simplify to get:
x = -5y + 5
Invert x and y in this equation to get:
y = -5x + 5
That should be your inverse equation.
Replace y with h(x) to get:
h(x) = -5x + 5
You could have used any letter other than h(x). a(x), b(x), c(x) would have done just as well. I just chose h(x) arbitrarily because that letter wasn't used yet and it was the next letter in line in the alphabet.
g(x) is your equation.
h(x) is your inverse equation.
If h(x) is the inverse equation, you should be able to see that h(x) and g(x) are reflections about the line y = x.
Graph both of these equations plus equation of y = x to see if that's true.
Graph is shown below:
The line in the middle going from bottom left to top right is the line of y = x.
The line with the very steep slope going from top left to bottom right is the line of y = -5x + 5.
The line with the shallow slope going from top left to bottom right is the line of y = -(1/5)x + 1.
It's a little hard top see but they look like they're reflections about the line y = x (mirror images).
The other way to tell is because the inverse function undoes what the function does.
What this results in is that h(g(x)) = x
Here's how that works.
g(x) = -(1/5)*x + 1
Let x = any real number.
We'll choose 160.
Replace x in g(x) with 160 to get:
g(160) = -(1/5)*(160) + 1 which equals -31
h(x) = -5x + 5
Replace x in h(x) with (-31) that we just calculated for g(x) to get:
h(g(x)) = h(-31) = -5*-31 + 5 which becomes 155 + 5 which equals 160.
g(160) = -31
h(-31) = 160
h(x) undid what g(x) did which makes h(x) the inverse function of g(x).
You could also have shown that h(g(x)) = x by just solving the equations in x without substituting numbers for x.
Here's how you would have done that:
g(x) = -(1/5)*x + 1
h(x)= -5x + 5
h(g(x)) = -5 * (-(1/5)*x + 1) + 5
You replaced x in h(x) with g(x) and you replaced x in (-5x + 5) with (-(1/5)*x + 1)
Simplify by distributing the multiplication to get:
h(g(x)) = (-5 * (-1/5) * x) - (5 * 1) + 5
Since -5 * (-1/5) * x) = x, and - (5 * 1) = - 5, your equation becomes:
(
