Final answer:
The function f(x) = x^2 + 5x - 2 evaluates to 22 at x=3, transforms to x^2 - 5x - 2 for f(-x), to -x^2 - 5x + 2 for -f(x), to 9x^2 + 15x - 2 for f(3x), and requires simplification of the difference quotient for (f(x+h)-f(x))/h.
Step-by-step explanation:
For the quadratic function f(x) = x^2 + 5x - 2, we can perform several operations to find the values requested:
- (a) f(3): Replace x with 3 and evaluate the expression. f(3) = 3^2 + 5(3) - 2 = 9 + 15 - 2 = 22.
- (b) f(-x): Replace x with -x in the function. f(-x) = (-x)^2 + 5(-x) - 2 = x^2 - 5x - 2.
- (c) -f(x): Multiply the function by -1. -f(x) = -1*(x^2 + 5x - 2) = -x^2 - 5x + 2.
- (d) f(3x): Replace x with 3x in the function. f(3x) = (3x)^2 + 5(3x) - 2 = 9x^2 + 15x - 2.
- (e) (f(x+h)-f(x))/h (h ≠ 0): Evaluate the function at x+h and x, then subtract and divide by h. f(x+h) = (x+h)^2 + 5(x+h) - 2, and f(x) = x^2 + 5x - 2. The difference quotient will be (f(x+h) - f(x))/h = ((x+h)^2 + 5(x+h) - 2 - (x^2 + 5x - 2))/h. Expanding and simplifying will give the result.