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Question: If the function \(f:R\to R\) and \(g:R\to R\) are defined by \(f\left( x \right)=2x+3\) and \(g\left...

If the function f:R→Rf:R\to R and g:R→Rg:R\to R are defined by f(x)=2x+3f\left( x \right)=2x+3 and g(x)=x2+7g\left( x \right)={{x}^{2}}+7 , then the values of xx for which g\left\\{ f\left( x \right) \right\\}=8 are

  1. 0,-6
  2. -1,-2
  3. 1,-1
  4. 0, 6
  5. 0, 2
Explanation

Solution

Here in this question we have been asked to find the value of xx for which g\left\\{ f\left( x \right) \right\\}=8 given that f(x)=2x+3f\left( x \right)=2x+3 and g(x)=x2+7g\left( x \right)={{x}^{2}}+7 . We will substitute the value of f(x)f\left( x \right) in g\left\\{ f\left( x \right) \right\\}=8 and simplify it further for answering this question.

Complete step-by-step solution:
Now considering from the question we have been asked to find the value of xx for which g\left\\{ f\left( x \right) \right\\}=8 given that f(x)=2x+3f\left( x \right)=2x+3 and g(x)=x2+7g\left( x \right)={{x}^{2}}+7 .
Now we will consider g(x)=x2+7g\left( x \right)={{x}^{2}}+7 and use it in g\left\\{ f\left( x \right) \right\\}=8 . Therefore we can say that f2(x)+7=8{{f}^{2}}\left( x \right)+7=8 .
By simplifying it further we will have f2(x)=1{{f}^{2}}\left( x \right)=1 . Hence we need to find the values for which f(x)f\left( x \right) is −1-1 or 11 .
Now we will evaluate both the conditions one after the other.
By evaluating the first condition we will get as follows.
Hence we can say that
f(x)=1⇒2x+3=1 ⇒2x=−2 ⇒x=−1 \begin{aligned} & f\left( x \right)=1\Rightarrow 2x+3=1 \\\ & \Rightarrow 2x=-2 \\\ & \Rightarrow x=-1 \\\ \end{aligned} .
Now by evaluating the second condition similarly as the first one we will get the conclusion as follows.
Similarly
f(x)=−1⇒2x+3=−1 ⇒2x=−4 ⇒x=−2 \begin{aligned} & f\left( x \right)=-1\Rightarrow 2x+3=-1 \\\ & \Rightarrow 2x=-4 \\\ & \Rightarrow x=-2 \\\ \end{aligned} .
Therefore we can conclude when it is given that f(x)=2x+3f\left( x \right)=2x+3 and g(x)=x2+7g\left( x \right)={{x}^{2}}+7 then value of xx for which g\left\\{ f\left( x \right) \right\\}=8 is -1,-2.
Hence we will mark the option “2” as correct.

Note: In the process of answering questions of this type, we should be sure with the concepts of functions. This question can also be answered by substituting the given options in the given expression g\left\\{ f\left( x \right) \right\\}=8 . We can also say that
\begin{aligned} & g\left\\{ f\left( x \right) \right\\}={{\left( 2x+3 \right)}^{2}}+7 \\\ & \Rightarrow 4{{x}^{2}}+16+12x \\\ \end{aligned} .
Now by simplifying this quadratic expression also we can evaluate the values of xx which satisfy the expression g\left\\{ f\left( x \right) \right\\}=8 .