Given the function f(x) = 5 over 5x - 16, explain how to find the domain and range of the function and it's inverse.
the domain for all rational functions is all reals, except where the denominator is zero. So, the domain here is all reals except x = 16/5
There is a horizontal asymptote at y=0, and there is a vertical asymptote there, positive on one side and negative on the other, so the range is all reals except y=0.
To find the domain and range of a function, we need to analyze the restrictions on the input values (domain) and the output values (range) of the function.
1. Domain of f(x):
The domain consists of all possible values that x can take. In this case, the only restriction on the domain is that the denominator (5x - 16) cannot be equal to zero, as division by zero is undefined. Therefore, to find the domain of f(x), we need to solve the equation 5x - 16 ≠ 0.
Solving for x:
5x - 16 ≠ 0
5x ≠ 16
x ≠ 16/5
So, the domain of f(x) is all real numbers except x = 16/5.
2. Range of f(x):
To find the range of f(x), we observe that as x varies, the function f(x) will take on different values. However, since f(x) is a rational function, we can determine the range by analyzing the behavior of the function for both large positive and negative values of x.
As x approaches positive infinity, 5x - 16 approaches positive infinity. Consequently, f(x) approaches 5 divided by a very large positive number, resulting in f(x) approaching zero. As x approaches negative infinity, 5x - 16 approaches negative infinity, and f(x) approaches 5 divided by a very large negative number, which results in f(x) also approaching zero. Therefore, we can conclude that the range of f(x) is all real numbers except for zero.
Now, let's move on to finding the domain and range of the inverse function of f(x).
3. Inverse of f(x):
To find the inverse of f(x), we need to switch the roles of x and f(x) and solve for x. Let y represent f(x), then the equation becomes y = 5/(5x - 16).
To find the inverse, let's proceed with the following steps:
Step 1: Swap x and y.
x = 5/(5y - 16)
Step 2: Solve for y.
Multiply both sides by (5y - 16):
x(5y - 16) = 5
Distribute on the left:
5xy - 16x = 5
Step 3: Isolate y.
5xy = 5 + 16x
y = (5 + 16x)/(5x)
Now we have the expression for the inverse function.
4. Domain of the inverse:
To find the domain of the inverse function, we need to consider the restrictions on the values that x can take in the inverse function. In this case, since the denominator cannot be zero, we set 5x ≠ 0 and solve for x:
5x ≠ 0
x ≠ 0
So, the domain of the inverse function is all real numbers except x = 0.
5. Range of the inverse:
To determine the range of the inverse function, recall the range of the original function f(x) was found to be all real numbers except zero. Since the inverse function is obtained by swapping x and y, the range of the inverse function is the same as the domain of the original function.
Therefore, the range of the inverse function is all real numbers except y = 0.
In summary:
Domain of f(x): All real numbers except x = 16/5.
Range of f(x): All real numbers except 0.
Domain of the inverse function: All real numbers except x = 0.
Range of the inverse function: All real numbers except 0.