Question

Given the functions f (x)=\x] and g(x)= 2 x + 31 – 4 a. Determine the

Answer 1

Functions can be transformed through translation, dilation, etc.

• The changes are horizontal translations and vertical compression.
• The domain of f(x) is $(-\infty,\infty)$, while its range is $(0,\infty)$
• The domain of g(x) is $(-\infty,\infty)$, while its range is $(-4,\infty)$

We have:

$f(x) = |x|$

$g(x) = 2|x - 3| - 4$

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

First, f(x) was translated to the left by 3 units.

The rule is:

$(x,y) \to (x-3.y)$

So, we have:

$f'(x) = |x - 3|$

Next, the function is compressed vertically by 2

The rule is:

$(x,y) \to (x,2y)$

So, we have:

$f"(x) = 2|x - 3|$

Lastly, the function is translated down by 4 units.

The rule is:

$(x,y) \to (x, y - 4)$

So, we have:

$g(x) = 2|x - 3| - 4$

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 $(-\infty,\infty)$

The y values start from 0 and opens upward; so, its range is $(0,\infty)$

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 $(-\infty,\infty)$

The y values start from -4 and opens upward; so, its range is $(-4,\infty)$

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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