Problem 54
Question
Begin by graphing \(f(x)=\log _{2} x\). Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$ g(x)=\log _{2}(x+2) $$
Step-by-Step Solution
Verified Answer
The vertical asymptote is at \(x = -2\). The domain of the function \(g(x) = \log_2(x+2)\) is \((-\infty,-2)\) and the range is \((-\infty, \infty)\).
1Step 1: Graph the parent function f(x)
Firstly, sketch the graph of the function \(f(x) = \log_2 x\). The graph intersects the x-axis at (1,0), and there is a vertical asymptote at x=0.
2Step 2: Perform the transformation
Now perform the transformation to graph the function \(g(x) = \log_2(x+2)\). This function is shifted 2 units to the left compared to f(x), so the vertical asymptote is now at \(x = -2\) and the x-intercept is now at \(x = -1\).
3Step 3: Determine domain and range
From the graph, it can seen that the domain of g(x) is \((-\infty,-2)\) and the range is \((-\infty, \infty)\).
Key Concepts
Graph TransformationsVertical AsymptoteDomain and Range
Graph Transformations
When dealing with logarithmic functions, graph transformations are key in understanding how different functions relate to each other. The parent function in our exercise is \(f(x) = \log_2 x\). To graph \(g(x) = \log_2(x+2)\), we use a graph transformation technique known as horizontal shifting.
Horizontal shifts occur when we add or subtract a constant from the \(x\)-value inside the logarithm. In this case, the transformation is \(x + 2\), which indicates a shift 2 units to the left. This is because adding a number inside the function shifts the graph in the opposite direction to the sign of the number.
Transformations allow us to predict where key features of the graph, such as points and asymptotes, will move. Remember, left and right movements in graphs are always counter-intuitive: adding moves left, subtracting moves right.
Horizontal shifts occur when we add or subtract a constant from the \(x\)-value inside the logarithm. In this case, the transformation is \(x + 2\), which indicates a shift 2 units to the left. This is because adding a number inside the function shifts the graph in the opposite direction to the sign of the number.
Transformations allow us to predict where key features of the graph, such as points and asymptotes, will move. Remember, left and right movements in graphs are always counter-intuitive: adding moves left, subtracting moves right.
Vertical Asymptote
A vertical asymptote in a graph of a logarithmic function is a line which the graph approaches but never touches. This appears due to limitations within the function that make it undefined at certain points.
For the parent function \(f(x) = \log_2 x\), there is a vertical asymptote at \(x = 0\), because \(\log_2 x\) is undefined for \(x \leq 0\). As the graph is transformed for \(g(x) = \log_2(x+2)\), the asymptote moves from \(x=0\) to \(x=-2\). This occurs due to the horizontal shift transformation explained earlier.
Vertical asymptotes are crucial in shaping the graph of logarithmic functions and are directly responsible for defining the domain of these functions.
For the parent function \(f(x) = \log_2 x\), there is a vertical asymptote at \(x = 0\), because \(\log_2 x\) is undefined for \(x \leq 0\). As the graph is transformed for \(g(x) = \log_2(x+2)\), the asymptote moves from \(x=0\) to \(x=-2\). This occurs due to the horizontal shift transformation explained earlier.
Vertical asymptotes are crucial in shaping the graph of logarithmic functions and are directly responsible for defining the domain of these functions.
Domain and Range
Understanding the domain and range of logarithmic functions is essential for accurately interpreting their graphs. The domain of a function is the set of all possible \(x\)-values that the function accepts, while the range is the set of all possible \(y\)-values it can produce.
For the function \(g(x) = \log_2(x+2)\), the domain is \((-\infty, -2)\). This means \(x\) must be greater than \(-2\) to make the logarithm valid, as \(x = -2\) creates the vertical asymptote where the function is undefined. The range of logarithmic functions such as this is \((\infty, \infty)\), indicating that the function's output can be any real number.
Recognizing the domain and range helps us understand where the function exists and how it behaves over different intervals.
For the function \(g(x) = \log_2(x+2)\), the domain is \((-\infty, -2)\). This means \(x\) must be greater than \(-2\) to make the logarithm valid, as \(x = -2\) creates the vertical asymptote where the function is undefined. The range of logarithmic functions such as this is \((\infty, \infty)\), indicating that the function's output can be any real number.
Recognizing the domain and range helps us understand where the function exists and how it behaves over different intervals.
Other exercises in this chapter
Problem 53
Use the compound interest formulas \(A-P\left(1+\frac{r}{n}\right)^{n t}\) and \(A-P e^{n}\) to solve Exercises \(53-56 .\) Round answers to the nearest cent. F
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Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions.
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In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places.
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In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(
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