To find [tex]\( g(x) \)[/tex], the function obtained by shifting [tex]\( f(x) = \sqrt{x} \)[/tex] up by 6 units and left by 4 units, we need to perform the following transformations step-by-step:
1. Shift Left by 4 Units:
To shift the function [tex]\( f(x) \)[/tex] to the left by 4 units, we replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( x + 4 \)[/tex]. This gives us:
[tex]\[
f(x+4) = \sqrt{x+4}
\][/tex]
2. Shift Up by 6 Units:
Next, we need to shift [tex]\( f(x+4) \)[/tex] up by 6 units. To do this, we add 6 to the result of [tex]\( f(x+4) \)[/tex]. This gives us:
[tex]\[
g(x) = \sqrt{x+4} + 6
\][/tex]
Therefore, the function [tex]\( g(x) \)[/tex] that represents [tex]\( f(x) = \sqrt{x} \)[/tex] shifted up by 6 units and left by 4 units is:
[tex]\[
g(x) = \sqrt{x+4} + 6
\][/tex]
Now, let's evaluate [tex]\( g(x) \)[/tex] for several [tex]\( x \)[/tex] values to see the transformation's effect clearly.
Given [tex]\( x \)[/tex] values:
[tex]\[
x = -4, 0, 4
\][/tex]
1. When [tex]\( x = -4 \)[/tex]:
[tex]\[
g(-4) = \sqrt{-4 + 4} + 6 = \sqrt{0} + 6 = 0 + 6 = 6.0
\][/tex]
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = \sqrt{0 + 4} + 6 = \sqrt{4} + 6 = 2 + 6 = 8.0
\][/tex]
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[
g(4) = \sqrt{4 + 4} + 6 = \sqrt{8} + 6 = \sqrt{4 \cdot 2} + 6 = 2\sqrt{2} + 6 \approx 2 \cdot 1.414 + 6 = 2.828 + 6 = 8.82842712474619
\][/tex]
Thus, for the example [tex]\( x \)[/tex] values, the corresponding [tex]\( g(x) \)[/tex] values are:
[tex]\[
x = [-4, 0, 4]
\][/tex]
[tex]\[
g(x) = [6.0, 8.0, 8.82842712474619]
\][/tex]
The transformation of the function [tex]\( \sqrt{x} \)[/tex] to [tex]\( \sqrt{x+4} + 6 \)[/tex] is demonstrated by these calculated values.
