Question

How do you find $ \left( {gof} \right)\left( { - 2} \right) $ given $ f(x) = - 3x + 4 $ and $ g(x) = {x^2} $ ?

Answer 1

Hint: We have to find the value of a composite function which consists of two given functions $ f(x) $ and $ g(x) $ . To find the value of the composite function for a given value of the variable we have to first find the composite function using $ \left( {gof} \right)(x) = g[f(x)] $ . Then we can put the value of $ x $ as $ - 2 $ to get the resulting value.

Complete step-by-step answer:
We have to find the value of a composite function $ \left( {gof} \right) $ for the value of variable $ - 2 $ . We have two functions $ f(x) = - 3x + 4 $ and $ g(x) = {x^2} $ .
First we will find the composite function and then we will put the value of the variable as $ - 2 $ to get the value of the function.
From the definition of composite function we know that,
 $ \left( {gof} \right)(x) = g[f(x)] $
i.e. the function $ f(x) $ itself becomes the independent variable for the function $ g(x) $ . We insert the function $ f(x) $ into the function $ g(x) $ to get the resulting expression.
Thus, we can write,
 $
  \left( {gof} \right)(x) = g[f(x)] \\
   \Rightarrow \left( {gof} \right)(x) = g\left( { - 3x + 4} \right) \;
 $
Since, $ g(x) = {x^2} $ , when we use $ f(x) = - 3x + 4 $ as the input variable for $ g(x) $ we get $ g\left( { - 3x + 4} \right) = {\left( { - 3x + 4} \right)^2} $
 $ \Rightarrow \left( {gof} \right)(x) = {\left( { - 3x + 4} \right)^2} $
Thus, we get the composite function as $ \left( {gof} \right)(x) = {\left( { - 3x + 4} \right)^2} $ .
Now we have to find the value of the function for $ x = - 2 $ .
We can write,
 $
  \left( {gof} \right)(x) = {\left( { - 3x + 4} \right)^2} \\
   \Rightarrow \left( {gof} \right)( - 2) = {\left( { - 3 \times ( - 2) + 4} \right)^2} \\
   \Rightarrow \left( {gof} \right)( - 2) = {\left( {6 + 4} \right)^2} = {10^2} = 100 \;
 $
Hence, the value of $ \left( {gof} \right)( - 2) = 100 $ .
So, the correct answer is “100”.

Note: Composite function comprises two functions where the value of one function becomes the input variable for the other function. We have to take care while finding the composite function as it is not commutative, i.e. $ gof \ne fog $ for all cases. When we are finding $ gof $ we have to insert the function $ f(x) $ in the function $ g(x) $ .