1.consider the function
F(x)=log3x and g(x)=4log3( x+1 ) and answer the following questions

A. Describe each of the transformations required to transform the graph of f(x) to g(x)

B. Using the transformations you described above, draw a scetch of g(x). (On a graph)

C. Describe and give the equations for any asymptotes

D. List any x and y intercepts for any asymptotes and What is the behavior of g(x)?

The transformation from f(x) to g(x) is a horizontal shift 3 units left, followed by a vertical stretch by a factor of 4
The x and y intercepts are -0.67 and 1.91, respectively.
The behavior of g(x) is that, g(x) approaches infinity, as x approaches infinity.

The functions are given as:

[tex]\mathbf{f(x) = log3x}[/tex]

[tex]\mathbf{g(x) = 4log3(x+1)}[/tex]

(a) The transformation from f(x) to g(x)

First, f(x) is shifted left by 1 unit.

The rule of this transformation is:

[tex]\mathbf{(x,y) \to (x + 1,y)}[/tex]

So, we have:

[tex]\mathbf{f'(x) = log3(x + 1)}[/tex]

Next. f'(x) is vertically stretched by a factor of 4.

The rule of this transformation is:

[tex]\mathbf{(x,y) \to (x,4y)}[/tex]

So, we have:

[tex]\mathbf{g(x) = 4log3(x+1)}[/tex]

Hence, the transformation from f(x) to g(x) is a horizontal shift 3 units left, followed by a vertical stretch by a factor of 4

(b) Sketch of g(x)

See attachment

(c) Asymptotes

The graphs of g(x) have no asymptote

(d) The intercepts, and the behavior of f(x)

The graph crosses the x-axis at x =-0.67, and it crosses the y-axis at y = 1.91

Hence, the x and y intercepts are -0.67 and 1.91, respectively.

The behavior of g(x) is that, g(x) approaches infinity, as x approaches infinity.

We know this because, the value of the function increases as x increases

Read more about function transformations at:

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