Begin by graphing f(x)=log4x.

Question

Begin by graphing

f(x)=log4x.

Then use transformations of this graph to graph the given function. What is the vertical​ asymptote? Use the graphs to determine the given​ function's domain and range.

h(x)=4+log4x

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Part 1

Graph the function

h(x)=4+log4x.

Graph the asymptote of​ h(x) as a dashed line. Use the graphing tool to graph the function.

Part 2

What is the vertical​ asymptote?

enter your response here

​(Type an​ equation.)

Part 3

What is the domain of

h(x)=4+log4x​?

enter your response here

​(Simplify your answer. Type your answer in interval​notation.)

Part 4

What is the range of

h(x)=4+log4x​?

enter your response here

​(Simplify your answer. Type your answer in interval​notation.)

Answer 1

Part 1 - Do this in a graphing tool

Part 2 - The vertical asymptote of the function log x is the vertical line x = 0. The vertical asymptote doesn't change for the function log 4x + 4. (This is the same as 4+log4x, written in standard form.)

Part 3 - The domain of the function log x is 0 < x < infinity. (You can't take the log of zero or a negative number.) The domain doesn't change for the function log 4x + 4.

Part 4 - The range of the function log x is negative infinity < y < infinity. The range doesn't change for the function log 4x + 4.

In case you are wondering how functions log x and log 4x + 4 are different, here is a summary of transformations:

The general rules of transformations for the function of the format f(b(x+h)) + k are

k - vertical shift up/down

h - horizontal shift down/up (the opposite) - So positive h=3 means a horizontal shift left (negative) 3 spaces.

a - vertical stretch/shrink. Negative a is reflection over x-axis.

b - horizontal shrink/stretch (the opposite) - So b=3 is a horizontal shrink of 1/3. Negative b is reflection over y-axis. Also, note the parentheses b(x+h), b distributes to both the x and the h.

So the function log 4x + 4 is the function log x squished horizontally by 1/4 and shifted upward by 4.