Final answer:
To find f(5/4), substitute g(x) into the equation (gof)x = 4x² - 10x + 5 and solve for f(5/4). The solution is f(5/4) = -5.
Step-by-step explanation:
To find f(5/4), we need to solve for f(x) in the equation (gof)x = 4x² - 10x + 5.
Substitute g(x) into the equation:
(x²x+1)(f(x)) = 4x² - 10x + 5
Expand and simplify:
x²(f(x)) + (f(x)) = 4x² - 10x + 5
Using the identity property of multiplication, we know that x²(f(x)) = x²f(x). Rewriting the equation, we get:
x²f(x) + f(x) = 4x² - 10x + 5
Factor out f(x) from both terms:
f(x)(x² + 1) = 4x² - 10x + 5
Now, substitute x = 5/4 and solve for f(5/4):
f(5/4)(5/4² + 1) = 4(5/4)² - 10(5/4) + 5
f(5/4)(25/16 + 1) = 4(25/16) - 10(5/4) + 5
f(5/4)(25/16 + 16/16) = 4(25/16) - 10(5/4) + 5
f(5/4)(41/16) = 25/4 - 50/4 + 5
f(5/4)(41/16) = -20/4 + 5
f(5/4)(41/16) = -5
Therefore, f(5/4) = -5.