To solve for [tex]\( f[g(3)] \)[/tex], we need to follow these steps:
1. Determine [tex]\( g(3) \)[/tex]:
Given the function [tex]\( g(x) = 5x - 2 \)[/tex], we substitute [tex]\( x = 3 \)[/tex] into the function:
[tex]\[
g(3) = 5 \cdot 3 - 2
\][/tex]
Simplifying this:
[tex]\[
g(3) = 15 - 2 = 13
\][/tex]
Therefore, [tex]\( g(3) = 13 \)[/tex].
2. Calculate [tex]\( f[g(3)] \)[/tex] by substituting [tex]\( g(3) \)[/tex] into [tex]\( f(x) \)[/tex]:
Given the function [tex]\( f(x) = 4x^2 - 3 \)[/tex], we now need to substitute [tex]\( x = 13 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(13) = 4 \cdot (13)^2 - 3
\][/tex]
First, compute [tex]\( 13^2 \)[/tex]:
[tex]\[
13^2 = 169
\][/tex]
Next, multiply by 4:
[tex]\[
4 \cdot 169 = 676
\][/tex]
Finally, subtract 3:
[tex]\[
676 - 3 = 673
\][/tex]
Therefore, [tex]\( f(13) = 673 \)[/tex].
3. Conclusion:
We have determined that [tex]\( f[g(3)] \)[/tex] is 673.
Thus, the correct answer is:
[tex]\[
f[g(3)] = 673
\][/tex]
