use the given logarithmic function to answer the following questions.

\f(x) = \ln(x - 4) - 2\

a) what are the domain and range of $f$?

domain

range

b) what is the $x$-intercept of the graph of $f$? enter your answer as an ordered pair. if no $x$-intercept exists, enter none.

$x$-intercept

c) the graph of $\ln(x)$ is shown below. sketch the graph of $f(x) = \ln(x - 4) - 2$ by entering values for the horizontal/vertical shift(s) in the boxes below, then clicking \graph\.

Question

use the given logarithmic function to answer the following questions.

\f(x) = \ln(x - 4) - 2\

a) what are the domain and range of $f$?

domain

range

b) what is the $x$-intercept of the graph of $f$? enter your answer as an ordered pair. if no $x$-intercept exists, enter none.

$x$-intercept

c) the graph of $\ln(x)$ is shown below. sketch the graph of $f(x) = \ln(x - 4) - 2$ by entering values for the horizontal/vertical shift(s) in the boxes below, then clicking \graph\.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Logarithmic Domain", "Function Transformations" ], "new_concepts": [ "Logarithmic Range", "Logarithmic Intercepts" ], "current_concepts": [ "Logarithmic Domain", "Logarithmic Range", "Logarithmic Intercepts", "Function Transformations" ] } ## Find the domain and range of $f(x) = \ln(x-4) - 2$ $$ \begin{aligned} &x - 4 > 0 \implies x > 4 \implies \text{Domain: } (4, \infty)\ &\text{Range: } (-\infty, \infty) \end{aligned} $$ ## Find the $x$-intercept of $f(x)$ $$ \begin{aligned} &\ln(x-4) - 2 = 0\ &\ln(x-4) = 2\ &x-4 = e^2\ &x = e^2 + 4\ &\text{$x$-intercept: } (e^2 + 4, 0) \end{aligned} $$ ## Determine the horizontal and vertical shifts $$ \begin{aligned} &f(x) = \ln(x-4) - 2 \text{ relative to } y = \ln(x)\ &\text{Horizontal shift: } 4 \text{ units to the right}\ &\text{Vertical shift: } 2 \text{ units down} \end{aligned} $$ ## Question a Domain: $(4, \infty)$ Range: $(-\infty, \infty)$ ## Question b $x$-intercept: $(e^2 + 4, 0)$ ## Question c Horizontal shift: 4 units to the right Vertical shift: -2 units down (or 2 units down) { "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.log(x)", "latex": "\ln(x)" }, 0.01, 10 ], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "name": "y = ln(x)", "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "Math.log(x - 4) - 2", "latex": "\ln(x - 4) - 2" }, 4.01, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "f(x) = ln(x-4) - 2", "withLabel": true } }, { "type": "line", "params": [ [4, -10], [4, 10] ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1, "dash": 2, "name": "x = 4", "withLabel": true } }, { "type": "point", "params": [ [11.389, 0] ], "properties": { "name": "(e^2+4, 0)", "size": 4, "color": "#F28A55", "withLabel": true } } ], "timestamps": [] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Logarithmic Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Find the domain and range of \(f(x) = \ln(x-4) - 2\)

Find the \(x\)-intercept of \(f(x)\)

Determine the horizontal and vertical shifts

Answer:

Question a

Question b

Question c