Question

b) identify the domain, range, asymptote, x-intercept, and transformations of each function.
f(x): domain:
g(x): domain:

c) compare and contrast the end behavior of these functions.

Explanation:

Identify the parent function and equations

We analyze the two logarithmic curves shown in the graph.
The standard parent logarithmic function is \(y = \log_b(x)\) with a vertical asymptote at \(x = 0\) and an x-intercept at \((1, 0)\).
Looking at the graph:

• One curve (let's denote it as \(f(x)\)) has a vertical asymptote at \(x = 0\) (the y-axis) and passes through the x-intercept \((1, 0)\). This is the parent logarithmic function, likely \(f(x) = \log_b(x)\) or specifically \(f(x) = \ln(x)\) or \(f(x) = \log(x)\).
• The other curve (let's denote it as \(g(x)\)) is shifted horizontally to the right. Its vertical asymptote is at \(x = 2\), and its x-intercept is at \((3, 0)\). This represents a horizontal translation of the parent function by 2 units to the right, so \(g(x) = f(x - 2)\).

Determine domain and range of each function

For logarithmic functions of the form \(y = \log(x - h)\):

• The domain is restricted by the argument being strictly positive: \(x - h > 0 \implies x > h\).
• The range of any logarithmic function is all real numbers: \((-\infty, \infty)\).

For \(f(x)\):

• Vertical asymptote is at \(x = 0\).
• Domain: \(x > 0\) or \((0, \infty)\).
• Range: All real numbers, \((-\infty, \infty)\).

For \(g(x)\):

• Vertical asymptote is at \(x = 2\).
• Domain: \(x > 2\) or \((2, \infty)\).
• Range: All real numbers, \((-\infty, \infty)\).

Identify asymptotes, intercepts, and transformations

For \(f(x)\):

• Vertical Asymptote: \(x = 0\)
• x-intercept: \((1, 0)\)
• Transformations: None (it is the parent function).

For \(g(x)\):

• Vertical Asymptote: \(x = 2\)
• x-intercept: \((3, 0)\)
• Transformations: Shifted horizontally to the right by 2 units.

Compare and contrast end behavior

Logarithmic functions of the form \(y = \log(x - h)\) have the following end behavior:

• As \(x\) approaches the vertical asymptote from the right (\(x \to h^+\)), the function decreases without bound: \(y \to -\infty\).
• As \(x\) increases without bound (\(x \to \infty\)), the function increases without bound: \(y \to \infty\).

Comparing \(f(x)\) and \(g(x)\):

• Similarities: Both functions have the same right-end behavior. As \(x \to \infty\), both \(f(x) \to \infty\) and \(g(x) \to \infty\).
• Differences: Their left-end behaviors approach different vertical asymptotes. For \(f(x)\), as \(x \to 0^+\), \(f(x) \to -\infty\). For \(g(x)\), as \(x \to 2^+\), \(g(x) \to -\infty\).

Answer:

Question b

• \(f(x)\) (Parent Function):
• Domain: \(x > 0\) (or \((0, \infty)\))
• Range: All real numbers (or \((-\infty, \infty)\))
• Asymptote: Vertical asymptote at \(x = 0\)
• x-intercept: \((1, 0)\)
• Transformations: None (parent function)

• \(g(x)\) (Shifted Function):
• Domain: \(x > 2\) (or \((2, \infty)\))
• Range: All real numbers (or \((-\infty, \infty)\))
• Asymptote: Vertical asymptote at \(x = 2\)
• x-intercept: \((3, 0)\)
• Transformations: Shifted horizontally to the right by 2 units

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Question c

• Similarities: Both functions share the same right-end behavior: as \(x \to \infty\), both \(f(x) \to \infty\) and \(g(x) \to \infty\).
• Differences: Their left-end behaviors approach different vertical boundaries:
• For \(f(x)\), as \(x \to 0^+\), \(f(x) \to -\infty\).
• For \(g(x)\), as \(x \to 2^+\), \(g(x) \to -\infty\).