Function transformation involves changing the form of a function
- 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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