**Inverse of a quadratic function :**

The general form of a quadratic function is

f(x) = ax² + bx + c

Then, the inverse of the above quadratic function is

f⁻¹(x)

For example, let us consider the quadratic function

g(x) = x²

Then, the inverse of the quadratic function is g(x) = x² is

g⁻¹(x) = √x

We have to apply the following steps to find inverse of a quadratic function

**Step 1 :**

Let **f(x) **be a quadratic function

In the above function,

**f(x)** to be replaced by **"y" **or** y = f(x)**

So, y = quadratic function in terms of "x"

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

**Step 2 :**

Now, we have to redefine the function **y = f(x) **by **"x"** in terms of **"y"**

Then we will get **x = g(y)**

**Step 3 :**

In **x = g(y****)**, replace **"x"** by **f⁻¹(x)** and **"y"** by** "x"**.

Hence inverse of f(x) is, **f⁻¹(x) = g(x)**

**Example 1 :**

Find the inverse and its graph of the quadratic function given below.

f(x) = x²

**Solution :**

**Step 1 : **

In the given function, let us replace f(x) by "y". Then, we have

y = x²

**Step 2 :**

We have to redefine y = x² by "x" in terms of "y". Then we have

√y = x or x = √y

**Step 3 :**

In x = √y, replace "x" by f⁻¹(x) and "y" by "x".

Hence inverse of f(x) is

**f⁻¹(x) = √x **

**Graphing the inverse of f(x) :**

We can graph the original function by plotting the vertex (0, 0). The parabola opens up, because "a" is positive.

And we get f(1) = 1 and f(2) = 4, which are also the same values of f(-1) and f(-2) 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, (2, 4) becomes (4, 2).

We have to do this because the input value becomes the output value in the inverse, and vice versa.

The graph of the inverse is a reflection of the original function about the line y = x.

**Graph of f(x) and its inverse f⁻¹(x)**

**Example 1 :**

Find the inverse and its graph of the quadratic function given below.

f(x) = 2(x + 3)² - 4

**Solution :**

**Step 1 : **

In the given function, let us replace f(x) by "y". `Then, we have

y = 2(x + 3)² - 4

**Step 2 :**

We have to redefine y = x² by "x" in terms of "y". Then we have

**Step 3 :**

Replacing "x" by f⁻¹(x) and "y" by "x" in the last step, we get inverse of f(x)

Hence inverse of f(x) is

**Graphing the inverse of f(x) :**

We can graph the original function by taking (-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.

**Graph of f(x) and its inverse f⁻¹(x)**

After having gone through the stuff given above, we hope that the students would have understood "Inverse of a quadratic function".

Apart from the stuff given above, if you want to know more about "Inverse of a quadratic function", please click here

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**