To determine [tex]\( g(x) \)[/tex] when the function [tex]\( f(x) = x^2 + 5x - 1 \)[/tex] is shifted 3 units to the left, we need to understand how horizontal shifts affect the function.
### Step-by-Step Solution:
1. Horizontal Shift by [tex]\( h \)[/tex] Units:
A horizontal shift to the left by [tex]\( h \)[/tex] units involves replacing [tex]\( x \)[/tex] with [tex]\( x + h \)[/tex] in the function. For example, shifting [tex]\( f(x) \)[/tex] 3 units to the left requires us to replace [tex]\( x \)[/tex] with [tex]\( x + 3 \)[/tex].
2. Constructing [tex]\( g(x) \)[/tex]:
To find the new function [tex]\( g(x) \)[/tex] after the horizontal shift:
[tex]\[
g(x) = f(x + 3)
\][/tex]
3. Substitution:
Substitute [tex]\( x + 3 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
g(x) = f(x + 3) = (x + 3)^2 + 5(x + 3) - 1
\][/tex]
4. Expanding the Expression:
To understand this better, we expand the terms:
[tex]\[
(x + 3)^2 = x^2 + 6x + 9
\][/tex]
[tex]\[
5(x + 3) = 5x + 15
\][/tex]
So,
[tex]\[
g(x) = x^2 + 6x + 9 + 5x + 15 - 1
\][/tex]
5. Combining the Terms:
Combine all like terms:
[tex]\[
g(x) = x^2 + 6x + 5x + 9 + 15 - 1 = x^2 + 11x + 23
\][/tex]
6. Final Form:
Upon further inspection, the form given by the answer choices does not require final expansion and should be left in the form:
[tex]\[
g(x) = (x + 3)^2 + 5(x + 3) - 1
\][/tex]
### Conclusion:
The correct function for [tex]\( g(x) \)[/tex] after shifting the original function [tex]\( f(x) \)[/tex] 3 units to the left is:
[tex]\[
g(x) = (x+3)^2 + 5(x+3) - 1
\][/tex]
Thus, the right answer is:
[tex]\[
\boxed{C: \; g(x) = (x+3)^2 + 5(x+3) - 1}
\][/tex]
