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  • If  and  <img alt="(gof)(x)=4x^{2}-10x+5," src="https://entrancecorner.oncodecogs.com/gif.latex?%28gof%29%28x%29%3

If g(x)=x^{2}+x-1 and  (gof)(x)=4x^{2}-10x+5, then f\left ( \frac{5}{4} \right ) is equal to:
Option: 1 -\frac{3}{2}
Option: 2 -\frac{1}{2}
Option: 3 \frac{1}{2}
Option: 4 \frac{3}{2}
 

Answers (1)

best_answer

\begin{align*} g(x)&=x^2+x-1 \\ gof(x)&= 4x^2-10x+5\\ f(x)&= ax+b\\ gof(x)&= (ax+b)^2+ax+b-1\\ gof(x)&= a^2x^2+b^2+2axb+ax+b-1 = 4x^2-10x+5\\ \implies a^2x^2&+x(2ab+a)+b^2+b-1=4x^2-10x+5\\ \end{align*}

Therefore , a^2=4, \ 2ab+a = -10, \ b^2+b-1=5

Solving the above equations, a=2, \ b = -3,

Therefore,

 f(x)=2x-3 \\ f\left(\frac{5}{4}\right)= 2 \times \frac{5}{4}-3

                 =\frac{-1}{2} 

Correct option (2)

Posted by

Kuldeep Maurya

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