Final answer:
The expression for (f(x+h)-f(x))/h for the function f(x) = 2x^2 - 3x + 4 simplifies to 4x + 2h - 3 after expansion and simplification, which is likely option D) 4x−3+2h considering a typo in the provided options.
Step-by-step explanation:
The question involves finding the expression for f(x+h)-f(x)/h for the given function f(x) = 2x^2 - 3x + 4. To solve for this, you first calculate f(x+h) using the original function with x substituted by (x+h), and then subtract f(x) from it. Finally, you divide the result by h.
To calculate f(x+h), you substitute (x+h) for every x in the function:
- f(x+h) = 2(x+h)^2 - 3(x+h) + 4
Then you expand this expression and simplify it:
- f(x+h) = 2x^2 + 4xh + 2h^2 - 3x - 3h + 4
By subtracting f(x) from f(x+h), you get:
- f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 - 3x - 3h + 4) - (2x^2 - 3x + 4)
After simplification:
- f(x+h) - f(x) = 4xh + 2h^2 - 3h
And dividing the entire expression by h:
- (f(x+h) - f(x))/h = (4xh + 2h^2 - 3h)/h = 4x + 2h - 3
This simplified expression matches none of the given options A) 2x−3+h B) 4x−3+h C) 2x−3+2h D) 4x−3+2h, but it's most likely due to a typo in the question. The correct answer should likely be D) 4x−3+2h.