Evaluate the difference quotient for the given function.
f(x)=1/x, f(x)-f(a)/(x-a).
Is the answer 1
(1/x - 1/a)/(x-a)
= [(a-x)/ax]/(x-a)
= -1(1/(ax))
= -1/(ax)
To evaluate the difference quotient for the function f(x) = 1/x, we need to find the expression f(x) - f(a) / (x - a).
First, let's calculate f(x) - f(a). Substituting the function values, we have:
f(x) - f(a) = 1/x - 1/a
Next, we divide the result by (x - a):
(f(x) - f(a)) / (x - a) = (1/x - 1/a) / (x - a)
To simplify further, we need to find a common denominator for 1/x and 1/a:
(a - x) / (x * a)
Thus, the difference quotient for f(x) = 1/x is (a - x) / (x * a).
To evaluate the difference quotient for the given function f(x) = 1/x, we need to find (f(x) - f(a))/(x - a).
First, let's find f(x) by substituting x into the function:
f(x) = 1/x
Next, let's find f(a) by substituting a into the function:
f(a) = 1/a
Now, let's substitute f(x) and f(a) into the difference quotient formula:
(f(x) - f(a))/(x - a) = (1/x - 1/a)/(x - a)
To simplify this expression, we need to find a common denominator for the fractions before subtracting:
Common denominator = x * a
Now, let's rewrite the expression with the common denominator:
[(1 * a - 1 * x)/(x * a)] / (x - a)
Simplifying the numerator, we get:
(a - x)/(x * a) / (x - a)
To divide by a fraction, we can multiply by its reciprocal. Therefore, to divide by (x - a), we can multiply by 1/(x - a):
(a - x)/(x * a) * 1/(x - a)
Now, we can cancel out the common factor of (x - a) in the numerator and the denominator:
-(x - a)/(x * a * (x - a))
Finally, we're left with:
-1/(x * a)
Hence, the evaluated difference quotient for the given function f(x)=1/x is -1/(x * a).