Question

Simplify the difference quotient (f(x)-f(a))/(x-a) for the given function. f(x)=2-5x-x^(2)

Answer 1

To simplify the difference quotient, substitute the function into the formula (f(x)-f(a))/(x-a), simplify the numerator by combining like terms, and factor out a negative sign. The simplified difference quotient for the given function f(x) = 2-5x-x^2 is (-5x + 5a + a^2 - x^2) / (x - a).

Step-by-step explanation:

To simplify the difference quotient, we substitute the given function f(x)=2-5x-x^2 into the formula (f(x)-f(a))/(x-a). First, we find f(x) and f(a) by plugging in the respective values of x and a into the function. Then, we substitute these values into the difference quotient formula and simplify the expression.

Let's simplify step by step:

1. Begin by finding f(x) and f(a): f(x) = 2 - 5x - x^2 and f(a) = 2 - 5a - a^2.
2. Substitute f(x) and f(a) into the difference quotient formula: [(2 - 5x - x^2) - (2 - 5a - a^2)] / (x - a).
3. Simplify the numerator: (2 - 5x - x^2 - 2 + 5a + a^2) / (x - a).
4. Combine like terms in the numerator: (-5x + 5a - x^2 + a^2) / (x - a).
5. Factor out a negative sign and group like terms: (-5x + 5a + a^2 - x^2) / (x - a).

So, the simplified difference quotient is (-5x + 5a + a^2 - x^2) / (x - a).