Question

The function $f$ satisfies the functional equation $3f\left( x \right) + 2f\left( {\dfrac{{x + 59}}{{x - 1}}} \right) = 10x + 30$ for all real $x \ne 1$. What is the value of $f\left( 7 \right)$?
A. 8
B. 4
C. -8
D. 11

Answer 1

Hint: First we will put $x = 7$ in the given equation. Again, we will put $x = 11$ in the equation. Then solve these two equations by using the elimination method. From the solutions, we can get the value of $f\left( 7 \right)$.

Formula Used:
The value of a function $f\left( x \right)$ at $x = a$ is $f\left( a \right)$.

Complete step by step solution:
Given function is $3f\left( x \right) + 2f\left( {\dfrac{{x + 59}}{{x - 1}}} \right) = 10x + 30$.
Substitute $x = 7$ in the given function:
$3f\left( 7 \right) + 2f\left( {\dfrac{{7 + 59}}{{7 - 1}}} \right) = 10 \cdot 7 + 30$
Simplify the above equation:
$ \Rightarrow 3f\left( 7 \right) + 2f\left( {\dfrac{{66}}{6}} \right) = 70 + 30$
$ \Rightarrow 3f\left( 7 \right) + 2f\left( {11} \right) = 100$ ……(i)

Now substitute $x = 11$ in the given function:
$3f\left( {11} \right) + 2f\left( {\dfrac{{11 + 59}}{{11 - 1}}} \right) = 10 \cdot 11 + 30$
Simplify the above equation:
$ \Rightarrow 3f\left( {11} \right) + 2f\left( {\dfrac{{70}}{{10}}} \right) = 110 + 30$
$ \Rightarrow 3f\left( {11} \right) + 2f\left( 7 \right) = 140$
$ \Rightarrow 2f\left( 7 \right) + 3f\left( {11} \right) = 140$ …..(ii)
Now we will solve the equation (i) and equation (ii) to get the value of $f\left( 7 \right)$.
Multiply equation (i) by 3 and equation (ii) by 2 and subtract them:
$\begin{array}{*{20}{c}}{9f\left( 7 \right)}& + &{6f\left( {11} \right)}& = &{300}\\{4f\left( 7 \right)}& + &{6f\left( {11} \right)}& = &{280}\end{array}$
$\overline {\begin{array}{*{20}{c}}{ - 5f\left( 7 \right)}&{}&{}& = &{ - 20}\end{array}} $
Divide both sides by $ - 5$:
$ \Rightarrow f\left( 7 \right) = \dfrac{{ - 20}}{{ - 5}}$
$ \Rightarrow f\left( 7 \right) = 4$

Option ‘B’ is correct

Additional information: A function is a relation with the input value to the output. The input value of a function is known as domain and the output value of the function is known as rage. The range of a function is also known as codomain. The domain and codomain are a set. If we get more than one output for an input, then the function is not a function.
 In the question, $x$ is an input and $f\left( x \right)$ is an output.

Note: This type of question should be solved by using the substitution method. So substitute $x=7$ in the given equation. The equation becomes the sum of two functions $f\left( 7 \right)$ and $f\left( 11 \right)$. Since we get another function $f\left( 11 \right)$, so substitute $x=11$ in the given equation. Then by solving these two equations, you will get the value of $f\left( 7 \right)$.