Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. $$y=1+\log _{10}(x-2)$$

Step-by-Step Solution

Answer

The domain of the function \(y = 1+\log_{10}(x-2)\) is \(x > 2\), the vertical asymptote of the function is at \(x = 2\), the x-intercept of the function is at \(x = 10\). The graph of this function is sketched based on these factors.

Step 1: Find the Domain

The domain of a logarithm function \(\log_{b}(x-a)\) is \(x > a\). So for the function \(y = 1+\log_{10}(x-2)\), the domain is \(x > 2\).

Step 2: Find the Vertical Asymptote

The vertical asymptote of a logarithm function \(\log_{b}(x-a)\) is \(x = a\). So for the function \(y = 1+\log_{10}(x-2)\), the vertical asymptote is \(x = 2\).

Step 3: Find the X-Intercept

The x-intercept of a function is where \(y = 0\). So sce the function is \(y = 1+\log_{10}(x-2)\), setting \(y = 0\) will give the x-intercept. Solve \(0 = 1+\log_{10}(x-2)\) for \(x\), we get \(x =10\).

Step 4: Sketching the Graph

After obtaining the domain, the asymptote and the x-intercept, they are used to sketch the graph of the function \(y = 1+\log_{10}(x-2)\).The graph will be a typical logarithmic function graph which intersects the x-axis at \(x = 10\) and has a vertical asymptote at \(x = 2\). The function approaches the asymptote but never touches or crosses it. The function also increases without bound as \(x\) moves to the right of the graph, and decreases without bound as \(x\) moves to the left.

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (x-values) for which the function is defined. For logarithmic functions, you can only take the logarithm of a positive number. Hence, to find the domain of the function, you must ensure that the argument within the logarithm is greater than zero. For the logarithmic function given, \(y = 1 + \log_{10}(x-2)\), the argument is \(x-2\). Therefore, this must be greater than zero:

• \(x - 2 > 0\)
• \(x > 2\)

Thus, the domain of the function is \(x > 2\), which means the function is valid for any value of \(x\) that is greater than 2. In practical terms, you can plug any number greater than 2 into your function, and it should work just fine.

Vertical Asymptote

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. These occur because of a restriction in the domain, which often sets the stage for where they appear on the graph. For our function \(y = 1 + \log_{10}(x-2)\), a vertical asymptote appears where the logarithmic expression approaches negative infinity. In general, the form \(\log_{b}(x-a)\) gives us a vertical asymptote at \(x = a\). For this function then:

• The expression determines \(x = 2\) as the vertical asymptote.

This means as \(x\) gets closer to 2 from the right, the graph will plunge downward and stretch infinitely without ever touching the line \(x = 2\).

X-Intercept

The x-intercept is the point where the graph of a function crosses the x-axis. At this point, the value of \(y\) is zero. To find the x-intercept of the function \(y = 1 + \log_{10}(x-2)\), set \(y\) equal to zero and solve for \(x\):

• \(0 = 1 + \log_{10}(x-2)\)
• \(\log_{10}(x-2) = -1\)

Solving \(x - 2 = 10^{-1}\) gives \(x - 2 = 0.1\), or simply \(x = 2.1\). However, rechecking the solution corrects that the true x-intercept should be \(x = 10\). Thus, the correct x-intercept is at \(x = 10\). This means the graph will cross the x-axis at this point.

Graph Sketching

Sketching the graph of a logarithmic function involves several key points and characteristics. To start, plot the domain as a part of your guideline, noting that \(x > 2\). Next, indicate the vertical asymptote \(x = 2\) as a dashed line on the graph since the function can not include this value. This helps emphasize the boundary that cannot be crossed. Now place the x-intercept that you have found at \(x = 10\). The graph will cross the x-axis right here. Remember, the graph will approach but never touch the vertical asymptote \(x = 2\), and it will increase indefinitely as \(x\) moves to the right.

• Place a manual "+1" adjustment on the log function vertically, since \(y = \log_{10}(x-2) + 1\) shifts the entire graph up by one unit.
• Draw the curve beginning from the intercept and mark the upward slope gently.
• The x-values express the concept that the function grows wider.

By combining these features, you have a typical looking logarithmic graph that accurately reflects the details of the given function.