Question

Given the function () = log2( + 4) − 2 :a. On a sheet of graph paper, use transformations to graph the function. Show theasymptote on your graph using a dashed or dotted line and write its equation.b. State the domain and range of this function.c. Algebraically calculate the x-intercept of the graph of this function.

Answer 1

a)

b)

`Domain:(-4,\infty)`
`Range:(-\infty,\infty)`

c) x-intercept is 0

Step-by-step explanation:

Given:

`f(x)=\log_2(x+4)-2`

a) See below the graph of different transformations of the given function;

The equation of the vertical asymptote as shown on the graph is x = -4

b) The domain of a function is the set of possible input values for which the function is defined. The domain of a graph is the set of possible values from left to right.

Looking at the given graph, we can see that the domain of the function is;

`Domain:(-4,\infty)`

The range of a graph is the set of values from the bottom to the top of the graph. Looking at the graph, we can see that the range is;

`Range:(-\infty,\infty)`

c) We'll follow the below steps to determine the x-intercept of the function;

Step 1: Substitute f(x) with 0;

`0=\log_2(x+4)-2`

Step 2: Add 2 to both sides;

`\begin{gathered} 0+2=\log_2(x+4)-2+2 \\ 2=\log_2(x+4) \end{gathered}`

Step 3: Apply the below rule;

`\begin{gathered} \log_ab=c \\ b=a^c \end{gathered}`
`\begin{gathered} 2^2=x+4 \\ 4=x+4 \\ 4-4=x \\ 0=x \\ \therefore x=0 \end{gathered}`

So the x-intercept is 0