To determine the [tex]$y$[/tex]-values of the two functions [tex]\( f(x) = 3^x - 3 \)[/tex] and [tex]\( g(x) = 7x^2 - 3 \)[/tex], let's analyze each function in detail:
### Function [tex]\( f(x) = 3^x - 3 \)[/tex]
1. Nature of the function:
[tex]\( f(x) = 3^x - 3 \)[/tex] is an exponential function shifted downward by 3 units.
2. Behavior as [tex]\( x \to -\infty \)[/tex]:
As [tex]\( x \)[/tex] goes to negative infinity, [tex]\( 3^x \)[/tex] approaches 0. Therefore, [tex]\( f(x) \)[/tex] approaches:
[tex]\[
f(x) = 3^x - 3 \to 0 - 3 = -3
\][/tex]
This implies that the smallest [tex]\( y \)[/tex]-value of [tex]\( f(x) \)[/tex] gets closer and closer to, but never actually reaches, [tex]\(-3\)[/tex].
3. Range of [tex]\( f(x) \)[/tex]:
Based on the above, the range of [tex]\( f(x) \)[/tex] is [tex]\(( -3, \infty )\)[/tex].
### Function [tex]\( g(x) = 7x^2 - 3 \)[/tex]
1. Nature of the function:
[tex]\( g(x) = 7x^2 - 3 \)[/tex] is a quadratic function opening upwards (since the coefficient of [tex]\( x^2 \)[/tex] is positive).
2. Vertex of the parabola:
The vertex of this parabola represents the minimum [tex]\( y \)[/tex]-value. For a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], the minimum value occurs at:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
For [tex]\( g(x) = 7x^2 - 3 \)[/tex], [tex]\( a = 7 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = -3 \)[/tex]:
[tex]\[
x = -\frac{0}{2 \cdot 7} = 0
\][/tex]
3. Minimum [tex]\( y \)[/tex]-value:
At [tex]\( x = 0 \)[/tex],
[tex]\[
g(0) = 7(0)^2 - 3 = -3
\][/tex]
This means the minimum [tex]\( y \)[/tex]-value of [tex]\( g(x) \)[/tex] is [tex]\(-3\)[/tex].
### Summary and Answer Choices Analysis
Looking at both functions, we summarize the findings:
- The minimum [tex]\( y \)[/tex]-value of [tex]\( f(x) \)[/tex] is [tex]\(-3\)[/tex], but [tex]\( f(x) \)[/tex] never actually reaches [tex]\(-3\)[/tex]; it approaches it asymptotically.
- The minimum [tex]\( y \)[/tex]-value of [tex]\( g(x) \)[/tex] is [tex]\(-3\)[/tex] at [tex]\( x = 0 \)[/tex].
Given these results, let's evaluate the answer choices:
A. [tex]$f(x)$[/tex] has the smallest possible [tex]$y$[/tex]-value.
This is False. While [tex]\( f(x) \)[/tex] approaches [tex]\(-3\)[/tex], it never actually reaches it.
B. The minimum [tex]\( y \)[/tex]-value of [tex]\( g(x) \)[/tex] is [tex]\(-3\)[/tex].
This is True. [tex]\( g(x) \)[/tex] has a minimum [tex]\( y \)[/tex]-value of [tex]\(-3\)[/tex] at [tex]\( x = 0 \)[/tex].
C. The minimum [tex]\( y \)[/tex]-value of [tex]\( f(x) \)[/tex] is [tex]\(-3\)[/tex].
This is True. The minimum [tex]\( y \)[/tex]-value that [tex]\( f(x) \)[/tex] approaches is indeed [tex]\(-3\)[/tex].
D. [tex]$g(x)$[/tex] has the smallest possible [tex]\( y \)[/tex]-value.
This is False since [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex] both approach a minimum [tex]\( y \)[/tex]-value of [tex]\(-3\)[/tex].
### Conclusion
Analyzing the [tex]$y$[/tex]-values of both functions leads us to the following correct choices:
- B and C are correct.
