Suppose that f(x) = 1/x. Find a simplified expression for f(x)-f(a)/x-a
f(x) = 1/x
so, f(a) = 1/a
(f(x) - f(a))/(x-a) , notice the necessary placing of brackets
= (1/x - 1/a)/(x-a)
= [ (a-x)/(ax) ]/(x-a)
= (a-x)/(ax) * 1/(x-a)
= -1/(ax)
notice (a-x)/(x-a)
= -(x-a)/(x-a) = -1
got it, thanks!
To find a simplified expression for (f(x) - f(a))/(x - a), we can start by substituting the expression for f(x) and f(a) into the numerator.
First, let's find f(x):
f(x) = 1/x
Next, let's find f(a):
f(a) = 1/a
Now, we can substitute these values into the numerator:
(f(x) - f(a)) = (1/x) - (1/a)
To simplify this expression, we need to find a common denominator for (x) and (a).
The common denominator is (x * a).
Now, let's rewrite (1/x) and (1/a) using the common denominator:
(1/x) = (a/(a * x))
(1/a) = (x/(x * a))
Substituting these into the numerator, we get:
(f(x) - f(a)) = (1/x) - (1/a) = (a/(a * x)) - (x/(x * a))
Now, we can combine the fractions in the numerator:
(f(x) - f(a)) = (a - x)/(a * x)
Finally, we divide this numerator expression by (x - a):
((f(x) - f(a))/(x - a)) = (a - x)/(a * x) ÷ (x - a)
Dividing by (x - a) is the same as multiplying by the reciprocal:
((f(x) - f(a))/(x - a)) = (a - x)/(a * x) * (1/(x - a))
Now, we can simplify this expression by canceling out the common factor (x - a) in the numerator and denominator:
((f(x) - f(a))/(x - a)) = (a - x)/(a * x) * (1/(x - a)) = (-(x - a))/(a * x) = -(x - a)/(a * x)
Therefore, a simplified expression for (f(x) - f(a))/(x - a) is -(x - a)/(a * x).