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Chapter 3: Problem 78

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$f(x)=\ln (3-x)$$

Short Answer

Expert verified
The domain of the function is \(x < 3\), x-intercept is at \(x = 2\), and the function has a vertical asymptote at \(x = 3\).

Step by step solution

01

Find the domain of the function

The domain of a logarithmic function \(f(x) = \ln(x)\) are all real positive numbers. So for our function \(f(x)=\ln (3-x)\), the inside of the logarithm function (3 - x) must be greater than 0. Hence, we have \(3 - x > 0\). Solving this inequality for x, we get \(x < 3\). Therefore, the domain of the function is \(x < 3\).
02

Find the x-intercept

The x-intercept of the function is the value of x at which the function equals 0. So, we set \(f(x) = 0\), thus the equation becomes \(0 = \ln (3-x)\). From properties of logarithms, we know that this is true when \(3 - x = 1\). Solving this equation for x, we get \(x = 2\). So, the x-intercept is at \(x = 2\).
03

Find the vertical asymptote

The vertical asymptote of a logarithmic function is a vertical line at a value of x where the function approaches positive or negative infinity. From the domain of the function, we know that the function is not defined at \(x = 3\), thus x = 3 is the vertical asymptote of the function.
04

Sketch the graph

The graph of the function \(f(x) = \ln (3-x)\) should show the features determined in previous steps. The function has a vertical asymptote at \(x = 3\) and crosses the x-axis at x=2. Due to the minus sign in front of the x in the logarithm, the function will decrease from left to right.

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