4 multiple answer 1 point the logarithmic function ( g(x)=ln x ) is transformed to ( h(x)=ln (x + 2)-1 )…

Question 4

Explanation:

Step1: Analyze horizontal translation

For a function (y = f(x + c)), it is a horizontal translation. If (c>0), the graph of (y = f(x)) is translated (c) units to the left. For (h(x)=\ln(x + 2)-1) compared to (g(x)=\ln x), when (c = 2) in the (x) - part ((x\to x + 2)), (g(x)) is translated (2) units to the left.

Step2: Analyze vertical translation

For a function (y=f(x)-d) ((d>0)), the graph of (y = f(x)) is translated (d) units downward. For (h(x)=\ln(x + 2)-1) compared to (g(x)=\ln x), when (d = 1) (since (h(x)=g(x + 2)-1)), (g(x)) is translated (1) unit downward.

Step3: Analyze vertical asymptote

The vertical asymptote of (y=\ln x) is (x = 0). For (y=\ln(x + 2)-1), set (x+2=0), we get (x=-2). So the vertical asymptote of (y = g(x)) ((x = 0)) shifts (2) units to the left ((x=-2))

Answer:

• (g(x)) is translated (2) units to the left
• (g(x)) is translated (1) unit downward
• The vertical asymptote shifts (2) units to the left

Question 5

Explanation:

Step1: Find the inverse of (y = 5^{x + 1})

Start with (y = 5^{x+1}). Interchange (x) and (y): (x = 5^{y + 1})

Step2: Solve for (y)

Take the logarithm of both sides with base (5). Using the property (y=\log_{a}a^{z}=z) (if (a>0,a\neq1)), we have (\log_{5}x=\log_{5}(5^{y + 1})). By the power rule of logarithms (\log_{a}M^{n}=n\log_{a}M), (\log_{5}x=(y + 1)\log_{5}5). Since (\log_{5}5 = 1), we get (\log_{5}x=y + 1)

Step3: Isolate (y)

Subtract (1) from both sides: (y=\log_{5}x-1)

Answer:

(f^{-1}(x)=\log_{5}x - 1)