1. Find the inverse of the function f(x) = 2x + 3.
2. Find the inverse of the function f(x) = x2.
3. Find f-1(x) : f(x) = (x + 2)/(x - 2).
4. Find the inverse of the function h(x) = log10(x).
5. Find g-1(x) : g(x) = √(x - 5).
6. Find the inverse of the quadratic function and graph it.
f(x) = 2(x + 3)2 - 4
1. Answer :
f(x) = 2x + 3
Replace f(x) by y.
y = 2x + 3
Interchange x and y.
x = 2y + 3
x - 3 = 2y
Solve for y.
y = (x - 3)/2
Replace y by f-1(x).
f-1(x) = (x - 3)/2
f-1 (x) = (x - 3)/2
2. Answer :
Replace f(x) by y.
y = x2
Interchange x and y.
x = y2
y2 = x
Solve for y.
Take square root on both sides.
y = ±√x
Replace y by f-1(x).
f-1(x) = ±√x
3. Answer :
f(x) = (x + 2)/(x - 2)
Replace f(x) by y.
y = (x + 2)/(x - 2)
Interchange x and y.
x = (y + 2)/(y - 2)
Solve for y.
x = (y + 2)/(y - 2)
x(y - 2) = y + 2
xy - 2x = y + 2
xy - y = 2x + 2
y(x - 1) = 2x + 2
y = (2x + 2)/(x - 1)
Replace y by f-1(x).
f-1(x) = (2x + 2)/(x - 1)
4. Answer :
h(x) = log10(x)
Replace h(x) by y.
y = log10(x)
Interchange x and y.
x = log10(y)
Solve for y.
y = 10x
Replace y by h-1(x).
h-1(x) = 10x
5. Answer :
g(x) = √(x - 5)
Replace f(x) by y.
y = √(x - 5)
Interchange x and y.
x = √(y - 5)
Solve for y.
x2 = y - 5
y = x2 + 5
Replace y by f-1(x).
f-1(x) = x2 + 5
6. Answer :
Replace f(x) by y.
y = 2(x + 3)2 - 4
Interchange x and y.
x = 2(y + 3)2 - 4
Solve for y.
x + 4 = 2(y + 3)2
(x + 4)/2 = (y + 3)2
Take square root on both sides.
±√[(x + 4)/2] = y + 3
±√[(x + 4)/2] - 3 = y
y = -3 ± √[(x + 4)/2]
Replace y by f-1(x).
f-1(x) = -3 ± √[(x + 4)/2]
Graphing the inverse of f(x) :
We can graph the original function by plotting the vertex (-3, -4). The parabola opens up, because a is positive.
And we get f(-2) = -2 and f(-1) = 4, which are also the same values of f(-4) and f(-5) respectively.
To graph f-1(x), we have to take the coordinates of each point on the original graph and switch the x and y coordinates.
For example, (-1, 4) becomes (4, -1).
We have to do this because the input value becomes the output value in the inverse, and vice versa.
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