**Inverse Functions Worksheet : **

Worksheet given in this section will be much useful for the students who would like to practice problems on inverse functions.

Before look at the worksheet, if you would like to learn about inverse functions,

**Problem 1 :**

Find the inverse of the function :

f(x) = 2x + 3

**Problem 2 :**

Find f^{-1}(x) :

f(x) = (x + 2)/(x - 2)

**Problem 3 :**

Find the inverse of the function :

h(x) = log_{10}(x)

**Problem 4 :**

Find g^{-1}(x) :

g(x) = √(x - 5)

**Problem 1 :**

Find the inverse of the function :

f(x) = 2x + 3

**Solution : **

**Step 1 :**

Given function : f(x) = 2x + 3

In the above function f(x) to be replaced by y.

Then, we will get

y = 2x + 3

y = 2x + 3 has been defined by y in terms of x.

**Step 2 :**

Now we have to redefine y = 2x + 3 by x in terms of y.

y = 2x + 3

Subtract 3 from each side.

y - 3 = 2x

Divide each side by 2.

(y-3)/2 = x

x = (y-3) / 2

Now, the function has been defined by x in terms of y.

**Step 3 : **

In x = (y - 3)/2, replace x by f^{-1} (x) and y by x.

So, inverse of f(x) is,

f^{-1} (x) = (x - 3)/2

**Problem 2 :**

Find f^{-1}(x) :

f(x) = (x + 2)/(x - 2)

**Solution :**

**Step 1 :**

Given function : f(x) = (x + 2)/(x - 2).

In the above function f(x) to be replaced by y.

Then, we will get

y = (x + 2)/(x - 2)

y = (x + 2)/(x - 2) has been defined by y in terms of x.

**Step 2 :**

Now we have to redefine y = (x + 2)/(x - 2) by x in terms of y.

y = (x + 2)/(x - 2)

Multiply each side by (x - 2).

y(x - 2) = x + 2

xy - 2x = x + 2

Subtract x from each side.

xy - 3x = 2

x(y - 3) = 2

Divide each side by (y - 3).

x = 2/(y - 3)

Now, the function has been defined by x in terms of y.

**Step 3 :**

In x = 2/(y - 3) replace x by f^{-1} (x) and y by x.

So, inverse of f(x) is,

f^{-1} (x) = 2/(x - 3)

**Problem 3 : **

Find the inverse of the function :

h(x) = log_{10}(x)

**Solution : **

**Step 1 :**

Given function : h(x) = log₁₀(x)

In the above function h(x) to be replaced by y.

Then, we will get

y = log_{10}(x)

y = log_{10}(x) has been defined by y in terms of x.

**Step 2 :**

Now we have to redefine y = log_{10}(x) by x in terms of y.

y = log_{10}(x)

10^{y} = x

x = 10^{y}

Now, the function has been defined by x in terms of y.

**Step 3 :**

In x = 10^{y} replace x by h^{-1}(x) and y by x.

So, inverse of h(x) is,

h^{-1}(x) = 10^{x}

**Problem 4 :**

Find g^{-1}(x) :

g(x) = √(x - 5)

**Solution :**

**Step 1 :**

Given function : g(x) = √(x - 5).

In the above function f(x) to be replaced by y.

Then, we will get

g = √(x - 5)

y = √(x - 5) has been defined by y in terms of x.

**Step 2 :**

Now we have to redefine y = √(x - 5) by x in terms of y.

y = √(x - 5)

Square each side.

y^{2} = [√(x - 5)]^{2}

y^{2} = x - 5

Add 5 to each side.

y^{2} + 5 = x

x = y^{2} + 5

Now, the function has been defined by x in terms of y.

**Step 3 :**

In x = y^{2} + 5 replace x by g^{-1}(x) and y by x.

So, inverse of g(x) is,

g^{-1}(x) = x^{2} + 5

After having gone through the stuff given above, we hope that the students would have understood, how to find inverse of a function.

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