If g(x)=x²+x-2 and 1/2(g f)(x)=2x²-5x+2 then one such function f(x)=

Correct Answer is : 2x−32x-32x−3

Correct Answer is : 2x−32x-32x−3

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AP EAMCET Engineering 2018 Apr 23 Shift 2 Paper

Section: Mathematics

Question : 1 of 160

Marks: +1, -0

g(x)=x^{2}+x-2

\frac{1}{2}(g \circ f)(x)=2x^{2}-5x+2

f(x)=

Solution:

Consider the expression,

g(x)=x^{2}+x-2

And,

\frac{1}{2}(g \circ f)(x)=2x^{2}-5x+2

Then,

\frac{1}{2}(g \circ f)(x)=4x^{2}-10x+4

And,

(f(x))^{2}+f(x)-2=4x^{2}-10x+4

For the function to be linear

f(x)=ax+b

so,

(ax+b)^{2}+(ax+b)-2

=4x^{2}-10x+4

a^{2}x^{2}+(2ab+a)x+(b^{2}+b-2)

=4x^{2}-10x+4

By the comparison of coefficients,

a^{2}=4

And,

2ab+a=-10

And,

b^{2}+b-2=4

So,

a\pm 2

Then,

b=\begin{cases} -3 & \text{if } a=2 \\ 2 & \text{if } a=-2 \end{cases}

These values satisfies above relations, thus,

f(x)=2x-3 \text{ or } -2x+2

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