Answer :
To analyze the [tex]$y$[/tex]-values of the two functions [tex]\( f(x) = 3x^2 - 3 \)[/tex] and [tex]\( g(x) = 2^x - 3 \)[/tex], let's consider their behaviors and minimum points.
### Function [tex]\( f(x) = 3x^2 - 3 \)[/tex]
1. Form of the Function: This is a quadratic function in the standard form [tex]\( ax^2 + bx + c \)[/tex] with the leading coefficient [tex]\( a = 3 > 0 \)[/tex], indicating that it is a parabola opening upwards.
2. Finding Critical Points (Vertically opened parabola):
- The vertex of the parabola given by [tex]\( f(x) = 3x^2 - 3 \)[/tex] provides the minimum value of the function.
- For a quadratic [tex]\( ax^2 + bx + c \)[/tex], the vertex [tex]\( x \)[/tex] coordinate is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( b = 0 \)[/tex], so [tex]\( x = 0 \)[/tex].
3. Minimum [tex]\( y \)[/tex]-value:
- Substituting [tex]\( x = 0 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 3(0)^2 - 3 = -3 \][/tex]
- So, the minimum [tex]\( y \)[/tex]-value of [tex]\( f(x) \)[/tex] is [tex]\(-3\)[/tex].
### Function [tex]\( g(x) = 2^x - 3 \)[/tex]
1. Form of the Function: This function is an exponential function shifted downward by 3 units.
2. Behavior as [tex]\( x \to -\infty \)[/tex]:
- As [tex]\( x \to -\infty \)[/tex]:
- [tex]\( 2^x \to 0 \)[/tex]. Therefore, [tex]\( g(x) \to 0 - 3 = -3 \)[/tex].
- The function [tex]\( g(x) \)[/tex] asymptotically approaches [tex]\(-3\)[/tex] but does not reach [tex]\(-3\)[/tex] for any finite value of [tex]\( x \)[/tex].
3. Since [tex]\( g(x) \)[/tex] can get arbitrarily close to [tex]\(-3\)[/tex] but never actually reaches [tex]\(-3\)[/tex], its minimum [tex]\( y \)[/tex]-value in practical terms is indefinitely close to [tex]\(-3\)[/tex].
### Conclusion
Based on the analyses above:
- The minimum [tex]\( y \)[/tex]-value of [tex]\( f(x) \)[/tex] is exactly [tex]\(-3\)[/tex].
- The function [tex]\( g(x) \)[/tex] approaches but never actually reaches a [tex]\( y \)[/tex]-value of [tex]\(-3\)[/tex].
Option B states that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have equivalent minimum [tex]\( y \)[/tex]-values, as both can reach or approach [tex]\(-3\)[/tex] equivalently in the context of their mathematical definitions and behaviors:
Thus,
[tex]\[ \boxed{B. \, f(x) \text{ and } g(x) \text{ have equivalent minimum } y\text{-values.}} \][/tex]
### Function [tex]\( f(x) = 3x^2 - 3 \)[/tex]
1. Form of the Function: This is a quadratic function in the standard form [tex]\( ax^2 + bx + c \)[/tex] with the leading coefficient [tex]\( a = 3 > 0 \)[/tex], indicating that it is a parabola opening upwards.
2. Finding Critical Points (Vertically opened parabola):
- The vertex of the parabola given by [tex]\( f(x) = 3x^2 - 3 \)[/tex] provides the minimum value of the function.
- For a quadratic [tex]\( ax^2 + bx + c \)[/tex], the vertex [tex]\( x \)[/tex] coordinate is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( b = 0 \)[/tex], so [tex]\( x = 0 \)[/tex].
3. Minimum [tex]\( y \)[/tex]-value:
- Substituting [tex]\( x = 0 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 3(0)^2 - 3 = -3 \][/tex]
- So, the minimum [tex]\( y \)[/tex]-value of [tex]\( f(x) \)[/tex] is [tex]\(-3\)[/tex].
### Function [tex]\( g(x) = 2^x - 3 \)[/tex]
1. Form of the Function: This function is an exponential function shifted downward by 3 units.
2. Behavior as [tex]\( x \to -\infty \)[/tex]:
- As [tex]\( x \to -\infty \)[/tex]:
- [tex]\( 2^x \to 0 \)[/tex]. Therefore, [tex]\( g(x) \to 0 - 3 = -3 \)[/tex].
- The function [tex]\( g(x) \)[/tex] asymptotically approaches [tex]\(-3\)[/tex] but does not reach [tex]\(-3\)[/tex] for any finite value of [tex]\( x \)[/tex].
3. Since [tex]\( g(x) \)[/tex] can get arbitrarily close to [tex]\(-3\)[/tex] but never actually reaches [tex]\(-3\)[/tex], its minimum [tex]\( y \)[/tex]-value in practical terms is indefinitely close to [tex]\(-3\)[/tex].
### Conclusion
Based on the analyses above:
- The minimum [tex]\( y \)[/tex]-value of [tex]\( f(x) \)[/tex] is exactly [tex]\(-3\)[/tex].
- The function [tex]\( g(x) \)[/tex] approaches but never actually reaches a [tex]\( y \)[/tex]-value of [tex]\(-3\)[/tex].
Option B states that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have equivalent minimum [tex]\( y \)[/tex]-values, as both can reach or approach [tex]\(-3\)[/tex] equivalently in the context of their mathematical definitions and behaviors:
Thus,
[tex]\[ \boxed{B. \, f(x) \text{ and } g(x) \text{ have equivalent minimum } y\text{-values.}} \][/tex]
