Given:
The function f(x) = ln(x) and g(x) = ln(x + 6).
Required:
Describe how a graph is transformation of the graph of f(x) = ln(x). Also, identify which attributes of f(x) = ln(x) change as a result of the transformation.
Explanation:
Let's first gather the required information:
Domain:
The domain of a function is the set of input values for f, in which the function is real and defined.
Range:
The range of a function comprises the set of values of a dependent variable for which the given function is defined.
End behavior:
The end behavior of a function describes the trend of the graph if we look to the right end of the x-axis (as x approcahes +∞) and to the left end of the x-axis (as x approaches −∞).
Vertical asymptote:
x - intercept:
To find the x-intercept, set y = 0 and solve for x.
Now, the graph of f(x) = ln(x) and g(x) = ln(x + 6).
So, graph of f(x) transformed 6 units left on x - axis as a g(x).
[tex]\begin{gathered} Domain: \\ f(x)=(0,\infty) \\ g(x)=(-6,\infty) \end{gathered}[/tex][tex]\begin{gathered} Range: \\ f(x)=(-\infty,\infty) \\ g(x)=(-\infty,\infty) \end{gathered}[/tex][tex]\begin{gathered} End\text{ }behavior: \\ f(x): \\ f(x)=ln(x)\rightarrow\infty\text{ as }x\rightarrow\infty\text{ }(ln(x)\text{ grows without bound as x grows} \\ \text{ without bound and }f(x)=ln(x)\rightarrow-\infty\text{ as }x\rightarrow0^{+\text{ }}\text{ }(ln(x)\text{ grows } \\ \text{ without bound in the negative direction as x approaches zero from the} \\ \text{ right\rparen.} \\ g(x): \\ as\text{ }x\rightarrow\infty,\text{ }f(x)\rightarrow+\infty \end{gathered}[/tex]
[tex]\begin{gathered} Vertical\text{ }asymptote: \\ f(x): \\ \text{ No vetical asymptotes.} \\ g(x): \\ \text{ No vertical asymptotes. } \end{gathered}[/tex][tex]\begin{gathered} X-intercept: \\ f(x): \\ (1,0) \\ g(x): \\ (-5,0) \end{gathered}[/tex][tex]\begin{gathered} \text{ Check where function is positive and negeative:} \\ f(x): \\ \text{ Function is negative for interval }(0,1)\text{ and positive in interval }(1,\infty). \\ g(x): \\ \text{ Function is negative for interval }(-6,-5)\text{ and positive for interval }(-5,\infty). \end{gathered}[/tex][tex]\begin{gathered} \text{ Check where function increases or decreases}:\text{ } \\ f(x): \\ \text{ Function increases for interval }(0,\infty). \\ g(x): \\ \text{ Function increases for interval }(-6,\infty). \end{gathered}[/tex]
Answer:
Completed answering the question.
