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Question: How do you write the inverse function for \(f\left( x \right)=\dfrac{1}{2}x+4\) ?...

How do you write the inverse function for f(x)=12x+4f\left( x \right)=\dfrac{1}{2}x+4 ?

Explanation

Solution

To find the inverse of the given function, first of all name f(x)f\left( x \right) as y in the above problem and then we will write x in terms of y. In the above problem, the function given is written in terms of x. After writing the x in terms of y then substitute y as x. And hence, we will get the inverse of the function.

Complete step by step solution:
The function given in the above problem of which we have to find the inverse of is as follows:
f(x)=12x+4f\left( x \right)=\dfrac{1}{2}x+4
Let us rename f(x)f\left( x \right) as y in the above equation and we get,
⇒y=12x+4\Rightarrow y=\dfrac{1}{2}x+4
As you can see that in the above equation, we have given y as a function in x so to find the inverse of the above function, we are going to rearrange the above equation in such a way so that we will get x as a function in y.
Subtracting 4 on both the sides we get,
⇒y−4=12x\Rightarrow y-4=\dfrac{1}{2}x
Multiplying 2 on both the sides we get,
⇒2(y−4)=12x(2)\Rightarrow 2\left( y-4 \right)=\dfrac{1}{2}x\left( 2 \right)
In the R.H.S of the above equation, 2 will be cancelled out in the numerator and denominator and we get,
⇒2(y−4)=x\Rightarrow 2\left( y-4 \right)=x
Now, multiplying 2 inside the bracket of the L.H.S of the above equation and we get,
⇒2y−8=x\Rightarrow 2y-8=x
Now, we are replacing xx as f−1(x){{f}^{-1}}\left( x \right) and y as xx in the above equation and we get,
⇒2x−8=f−1(x)\Rightarrow 2x-8={{f}^{-1}}\left( x \right)
Hence, we have found the inverse of the given function and is equal to 2x−82x-8.

Note: We can check whether the inverse of the function which we have calculated is correct or not by writing x in f−1(x){{f}^{-1}}\left( x \right) as f(x) and then see whether we are getting x or not.
Multiplying function with its inverse and we get,
f(x)=12x+4f\left( x \right)=\dfrac{1}{2}x+4
⇒f−1(x)=2x−8\Rightarrow {{f}^{-1}}\left( x \right)=2x-8
Substituting x as f(x) in the above inverse function we get,
⇒f−1(x)=2(f(x))−8 ⇒f−1(x)=2(12x+4)−8 \begin{aligned} & \Rightarrow {{f}^{-1}}\left( x \right)=2\left( f\left( x \right) \right)-8 \\\ & \Rightarrow {{f}^{-1}}\left( x \right)=2\left( \dfrac{1}{2}x+4 \right)-8 \\\ \end{aligned}
Multiplying 2 in the bracket of the above equation we get,
⇒f−1(x)=x+8−8 ⇒f−1(x)=x+0 ⇒f−1(x)=x \begin{aligned} & \Rightarrow {{f}^{-1}}\left( x \right)=x+8-8 \\\ & \Rightarrow {{f}^{-1}}\left( x \right)=x+0 \\\ & \Rightarrow {{f}^{-1}}\left( x \right)=x \\\ \end{aligned}
As we are getting x so the inverse of the function which we have found out is correct.