given f(x)=x-3 and g(x)=1/x-4 find the domain and range of the following functions

f(x)

f(g(x))

f(f(x))

I will assume you meant g(x) = 1/(x-4)

for f(x) = x-3, the domain and range are both the set of R

f(g(x))
= 1/(x-4) - 3

domain: any R, x≠4
range : any R, y ≠ -3

f(f(x) = (x-3) - 3
= x -6

domain: any R
range : any R

To find the domain and range of a function, we need to determine the set of all possible input (x-values) and output (y-values) of the function.

1. Domain and Range of f(x) = x - 3:

The domain is the set of all possible x-values that the function can take. In this case, there are no restrictions on the input x, so the domain of f(x) is all real numbers, or (-∞, +∞).

The range is the set of all possible output values that the function can produce. For f(x) = x - 3, as x varies from -∞ to +∞, the corresponding output values will also range from -∞ to +∞. Therefore, the range of f(x) is all real numbers, or (-∞, +∞).

Domain: (-∞, +∞)
Range: (-∞, +∞)

2. Domain and Range of f(g(x)):

To find the domain of f(g(x)), we need to consider the values that can be plugged into g(x) without resulting in any division by zero. The function g(x) = 1/(x - 4) is undefined when the denominator x - 4 equals zero. Therefore, x cannot be equal to 4. Thus, the domain of g(x) is all real numbers except 4, or (-∞, 4) U (4, +∞).

Now, we can substitute g(x) into f(x), which gives us: f(g(x)) = f(1/(x - 4)).
Since the domain of g(x) is (-∞, 4) U (4, +∞), we need to find the values of x for which 1/(x - 4) is defined within that domain. Since the denominator (x - 4) cannot be zero, the only restriction on the domain of f(g(x)) is that x should not be equal to 4.

Therefore, the domain of f(g(x)) is all real numbers except 4, which can be represented as (-∞, 4) U (4, +∞).

The range of f(g(x)) can be found by considering the possible values that f(g(x)) can take after evaluating the function for all valid x-values. Since f(x) has a range of all real numbers, the range of f(g(x)) will also be all real numbers.

Domain: (-∞, 4) U (4, +∞)
Range: (-∞, +∞)

3. Domain and Range of f(f(x)):

To find the domain of f(f(x)), we need to consider the values that can be used as input for f(f(x)).
Given f(x) = x - 3, any real number can be used as input for f(x), which means that any real number can be used as input for f(f(x)) as well. Therefore, the domain of f(f(x)) is (-∞, +∞).

For the range, we can evaluate f(f(x)) by applying f(x) twice as follows:

f(f(x)) = f(x - 3) = (x - 3) - 3 = x - 6.

Here, we have found that the output of f(f(x)) is x - 6, which implies that the range of f(f(x)) is also all real numbers, or (-∞, +∞).

Domain: (-∞, +∞)
Range: (-∞, +∞)