Question

If g(x)=x^2+x-1 and gof(x)=4x^2-10x+5 find f(5/4)

Answer 1

Therefore $f(\frac{5}{4} )=-\frac{1}{2}$

Step-by-step explanation:

To find the value of $f(\frac{5}{4} )$, we need to find out the function of f(x).

Let g(y)= gof(x)

The value of y in terms of x is the function of f(x)

Given that g(x)=x²+x-1

Then g(y)=y²+y-1 [ putting x=y]

Therefore,

g(y)= gof(x)

⇒y²+y-1 = 4x²-10x+5

⇒y²+y-1-4x²+10x-5=0

⇒y²+y -4x²+10x-6=0

$\Rightarrow y =\frac{-1\pm \sqrt{1^2-4.1.(-4x^2+10x-6)} }{2.1}$

$\Rightarrow y=\frac{-1\pm \sqrt{1+16x^2-40x+24} }{2}$

$\Rightarrow y=\frac{-1\pm \sqrt{16x^2-40x+25} }{2}$

$\Rightarrow y =\frac{-1\pm\sqrt{(4x-5)^2} }{2}$

$\Rightarrow y=\frac{-1\pm (4x-5)}{2}$

$\Rightarrow y=\frac{-1+4x-5}{2} ,\frac{-1-4x+5}{2}$

$\Rightarrow y=(2x-3) ,(-2x+2)$

Therefore either  f(x)=2x-3 or,  f(x)= -2x+2

When f(x)=2x-3

$f(\frac{5}{4} )=2.\frac{5}{4} -3=-\frac{1}{2}$

When f(x)= -2x+2

$f(\frac{5}{4} )=-2.\frac{5}{4} +2=-\frac{1}{2}$

Therefore $f(\frac{5}{4} )=-\frac{1}{2}$