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Question: What are the values of \( h \) and \( k \) ? Given \( f\left( x \right) = 4x + h \) and \( f'\left...

What are the values of hh and kk ?
Given f(x)=4x+hf\left( x \right) = 4x + h and f′(x)=5−kxf'\left( x \right) = 5 - kx , where hh and kk are constants. Find the values of hh and kk .

Explanation

Solution

First, we should analyze the data so that we are able to solve the given problem. Here we are given two functions: ff and the inverse of ff . We are asked to find the values of hh and kk where hh and kk are constants. Here, we need to convert the function ff into its inverse. Next, we shall compare the converted function and the given inverse function to obtain the desired answer.

Complete step by step answer:
We are given that f(x)=4x+hf\left( x \right) = 4x + h and f′(x)=5−kxf'\left( x \right) = 5 - kx , where hh and kk are constants. We need to find the values of hh and kk .
First, we shall convert the given f(x)=4x+hf\left( x \right) = 4x + h into its inverse function.
That means we shall find f−1(x){f^{ - 1}}\left( x \right) .
Let us consider f(x)=4x+hf\left( x \right) = 4x + h
f(x)−h=4xf\left( x \right) - h = 4x
⇒f(x)−h4=x\Rightarrow \dfrac{{f\left( x \right) - h}}{4} = x
⇒f(x)4−h4=x\Rightarrow \dfrac{{f\left( x \right)}}{4} - \dfrac{h}{4} = x
Now, we shall apply the inverse on both sides.
That means, f−1(f(x)14−h4)=f−1(x){f^{ - 1}}\left(f\left( x \right)\dfrac{1}{4} - \dfrac{h}{4}\right) = {f^{ - 1}}\left( x \right)
⇒x4−h4=f−1(x)\Rightarrow \dfrac{x}{4} - \dfrac{h}{4} = {f^{ - 1}}\left( x \right) (We know that f−1f(x)=x{f^{ - 1}}f\left( x \right) = x )
⇒f−1(x)=x4−h4\Rightarrow {f^{ - 1}}\left( x \right) = \dfrac{x}{4} - \dfrac{h}{4}
Thus, we obtained the required inverse of a function.
Now, we shall consider the given inverse f′(x)=5−kxf'\left( x \right) = 5 - kx
Let us compare both the inverse f−1(x)=x4−h4{f^{ - 1}}\left( x \right) = \dfrac{x}{4} - \dfrac{h}{4} and f′(x)=5−kxf'\left( x \right) = 5 - kx
Since both functions are the same, we shall compare the coefficient of xx
Thus, we have 14=−k\dfrac{1}{4} = - k
⇒−14=k\Rightarrow - \dfrac{1}{4} = k
Hence, k=−14k = - \dfrac{1}{4}
Similarly, we shall compare the constant term.
Thus, we get −h4=5- \dfrac{h}{4} = 5
⇒−h=5×4\Rightarrow - h = 5 \times 4
⇒−h=20\Rightarrow - h = 20
Hence, h=−20h = -20
Therefore, we have found the values of the constants h=−20h = -20 and k=−14k = - \dfrac{1}{4}

Note:
Students should note that we have converted the given function into its inverse. Then we compared the given inverse with the obtained inverse. We can also convert the given inverse of a function into its original function. Then we need to compare the given function with the obtained function. We can get the same answer as we got earlier.