Graph the function below. Determine the domain, range, and vertical asymptote. Please show all of your work.
f(x) = log2(x-4)
To graph the function f(x) = log2(x-4), we need to follow a step-by-step process. Here's how you can do it:
Step 1: Determine the Domain:
The domain of a logarithmic function is the set of all positive values that the function can take. In this case, since we have the expression (x-4) inside the logarithm, we must ensure that (x-4) is greater than zero.
So, we set (x-4) > 0 and solve for x:
x - 4 > 0
x > 4
Therefore, the domain of f(x) = log2(x-4) is x > 4.
Step 2: Determine the Range:
The range of a logarithmic function is the set of all possible values that the function's output can take. In this case, as x approaches positive infinity, the function values approach negative infinity.
So, the range of f(x) = log2(x-4) is (-∞, +∞).
Step 3: Determine the Vertical Asymptote:
A vertical asymptote occurs when the function approaches infinity or negative infinity as x approaches a specific value.
For a logarithmic function of the form f(x) = log2(x-4), the vertical asymptote occurs at the value where the argument of the logarithm, (x-4), is equal to zero.
So, we set (x-4) = 0 and solve for x:
x - 4 = 0
x = 4
Therefore, the vertical asymptote of f(x) = log2(x-4) is x = 4.
Step 4: Graph the function:
To graph the function, you can plot a few points and connect them smoothly.
Choose a few x-values greater than 4, such as x = 5, 6, and 7. Substitute these values of x into the function and compute the respective y-values:
For x = 5:
f(5) = log2(5-4) = log2(1) = 0
For x = 6:
f(6) = log2(6-4) = log2(2) = 1
For x = 7:
f(7) = log2(7-4) = log2(3) ≈ 1.585
Plot these points: (5, 0), (6, 1), and (7, 1.585).
Additionally, you can plot a point to the left of the vertical asymptote, such as x = 3:
For x = 3:
f(3) = log2(3-4)
Since the argument of the logarithm is negative, the function is undefined. Therefore, there is no point to the left of the vertical asymptote.
Finally, draw a smooth decreasing curve to the right of the vertical asymptote, passing through the plotted points.
The graph will approach but never reach the vertical asymptote x = 4.
So, to summarize:
- Domain: x > 4
- Range: (-∞, +∞)
- Vertical Asymptote: x = 4