Functions can be transformed through translation, dilation, etc.
- The changes are horizontal translations and vertical compression.
- The domain of f(x) is [tex](-\infty,\infty)[/tex], while its range is [tex](0,\infty)[/tex]
- The domain of g(x) is [tex](-\infty,\infty)[/tex], while its range is [tex](-4,\infty)[/tex]
We have:
[tex]f(x) = |x|[/tex]
[tex]g(x) = 2|x - 3| - 4[/tex]
(a) The changes from f(x) to g(x)
First, f(x) was translated to the left by 3 units.
The rule is:
[tex](x,y) \to (x-3.y)[/tex]
So, we have:
[tex]f'(x) = |x - 3|[/tex]
Next, the function is compressed vertically by 2
The rule is:
[tex](x,y) \to (x,2y)[/tex]
So, we have:
[tex]f"(x) = 2|x - 3|[/tex]
Lastly, the function is translated down by 4 units.
The rule is:
[tex](x,y) \to (x, y - 4)[/tex]
So, we have:
[tex]g(x) = 2|x - 3| - 4[/tex]
Hence, the changes are horizontal translations and vertical compression.
(b) Analyze f(x) and g(x)
f(x)
It spans across the x-axis; so, its domain is [tex](-\infty,\infty)[/tex]
The y values start from 0 and opens upward; so, its range is [tex](0,\infty)[/tex]
It crosses the x and y axes at 0; so its intercepts are 0.
g(x)
It spans across the x-axis; so, its domain is [tex](-\infty,\infty)[/tex]
The y values start from -4 and opens upward; so, its range is [tex](-4,\infty)[/tex]
It crosses the x-axis at -1 and -5; so its y-intercepts are -1 and -5
It crosses the y-axis at 2; so its x-intercept is 2
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