INVERSE OF A FUNCTION WORKSHEET

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

Answers

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 = 10^x

Replace y by h^-1(x).

h^-1(x) = 10^x

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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