Answer :
Let's analyze the [tex]$y$[/tex]-values of the two functions \( f(x) = 3^x - 3 \) and \( g(x) = 7x^2 - 3 \).
First, consider the function \( f(x) = 3^x - 3 \).
1. The function \( 3^x \) is an exponential function which is always positive for all real values of \( x \).
2. Subtracting 3 from \( 3^x \) shifts the entire graph of the exponential function down by 3 units.
3. As \( x \) approaches negative infinity, \( 3^x \) approaches 0 from the positive side. Thus:
[tex]\[ \lim_{x \to -\infty} (3^x - 3) = 0 - 3 = -3 \][/tex]
4. This means the minimum \( y \)-value that \( f(x) \) can approach is \( -3 \).
Next, consider the function \( g(x) = 7x^2 - 3 \).
1. The function \( 7x^2 \) is a parabola that opens upwards since the coefficient of \( x^2 \) is positive.
2. The minimum value of \( 7x^2 \) is 0, which occurs at \( x = 0 \).
3. Subtracting 3 from \( 7x^2 \) shifts the entire parabola down by 3 units.
4. Therefore, the minimum \( y \)-value of \( g(x) \) occurs at \( x = 0 \):
[tex]\[ g(0) = 7(0)^2 - 3 = -3 \][/tex]
We observe that both functions \( f(x) \) and \( g(x) \) have the minimum \( y \)-value of \(-3\).
Now, let's evaluate the given options:
A. The minimum \( y \)-value of \( f(x) \) is -3.
This is true, as we have seen that \( f(x) \) approaches -3 as \( x \) goes to negative infinity.
B. \( g(x) \) has the smallest possible \( y \)-value.
This is false. Both \( f(x) \) and \( g(x) \) have the same minimum \( y \)-value, which is -3.
C. The minimum \( y \)-value of \( g(x) \) is -3.
This is true, as we have seen that \( g(x) \) achieves -3 at \( x = 0 \).
D. \( f(x) \) has the smallest possible \( y \)-value.
This statement is also false. As established, both functions have the same minimum \( y \)-value of -3. Thus, \( f(x) \) does not have a smaller \( y \)-value than \( g(x) \).
Therefore, the correct answers are:
A. The minimum \( y \)-value of \( f(x) \) is -3.
C. The minimum [tex]\( y \)[/tex]-value of [tex]\( g(x) \)[/tex] is -3.
First, consider the function \( f(x) = 3^x - 3 \).
1. The function \( 3^x \) is an exponential function which is always positive for all real values of \( x \).
2. Subtracting 3 from \( 3^x \) shifts the entire graph of the exponential function down by 3 units.
3. As \( x \) approaches negative infinity, \( 3^x \) approaches 0 from the positive side. Thus:
[tex]\[ \lim_{x \to -\infty} (3^x - 3) = 0 - 3 = -3 \][/tex]
4. This means the minimum \( y \)-value that \( f(x) \) can approach is \( -3 \).
Next, consider the function \( g(x) = 7x^2 - 3 \).
1. The function \( 7x^2 \) is a parabola that opens upwards since the coefficient of \( x^2 \) is positive.
2. The minimum value of \( 7x^2 \) is 0, which occurs at \( x = 0 \).
3. Subtracting 3 from \( 7x^2 \) shifts the entire parabola down by 3 units.
4. Therefore, the minimum \( y \)-value of \( g(x) \) occurs at \( x = 0 \):
[tex]\[ g(0) = 7(0)^2 - 3 = -3 \][/tex]
We observe that both functions \( f(x) \) and \( g(x) \) have the minimum \( y \)-value of \(-3\).
Now, let's evaluate the given options:
A. The minimum \( y \)-value of \( f(x) \) is -3.
This is true, as we have seen that \( f(x) \) approaches -3 as \( x \) goes to negative infinity.
B. \( g(x) \) has the smallest possible \( y \)-value.
This is false. Both \( f(x) \) and \( g(x) \) have the same minimum \( y \)-value, which is -3.
C. The minimum \( y \)-value of \( g(x) \) is -3.
This is true, as we have seen that \( g(x) \) achieves -3 at \( x = 0 \).
D. \( f(x) \) has the smallest possible \( y \)-value.
This statement is also false. As established, both functions have the same minimum \( y \)-value of -3. Thus, \( f(x) \) does not have a smaller \( y \)-value than \( g(x) \).
Therefore, the correct answers are:
A. The minimum \( y \)-value of \( f(x) \) is -3.
C. The minimum [tex]\( y \)[/tex]-value of [tex]\( g(x) \)[/tex] is -3.
