f(x)=x-5 g(x)=1/x-1
find the range of the following
f(g(x))
g(f(x))
To find the range of a composite function, we first need to evaluate the composite function and then determine the range of the resulting expression.
1. Finding f(g(x)):
To evaluate f(g(x)), we substitute g(x) into the function f(x).
f(g(x)) = f(1/x - 1)
Now, substitute the expression for g(x) into the function f(x):
f(g(x)) = f(1/x - 1) = (1/x - 1) - 5
Simplifying:
f(g(x)) = 1/x - 1 - 5 = 1/x - 6
2. Finding g(f(x)):
To evaluate g(f(x)), we substitute f(x) into the function g(x).
g(f(x)) = g(x - 5)
Now, substitute the expression for f(x) into the function g(x):
g(f(x)) = g(x - 5) = 1/(x - 5) - 1
Simplifying:
g(f(x)) = 1/(x - 5) - 1
Now that we have the expressions for f(g(x)) and g(f(x)), let's find their respective ranges.
Range of f(g(x)) = 1/x - 6:
Since 1/x is a reciprocal function, it can take on any value except 0. Therefore, the range of 1/x is (-∞, 0) ∪ (0, +∞). Subtracting 6 from this range gives us the final range of f(g(x)).
Range(f(g(x))) = (-∞, -6) ∪ (6, +∞)
Range of g(f(x)) = 1/(x - 5) - 1:
Similar to 1/x, the function 1/(x - 5) also takes on any value except 0 since we are subtracting 5 from x. Therefore, the range is (-∞, 5) ∪ (5, +∞). Subtracting 1 from this range yields the final range of g(f(x)).
Range(g(f(x))) = (-∞, 4) ∪ (6, +∞)
Thus, the range of f(g(x)) is (-∞, -6) ∪ (6, +∞), and the range of g(f(x)) is (-∞, 4) ∪ (6, +∞).