Sure, let's find [tex]\( f(6t) \)[/tex] given the function [tex]\( f(x) = 3x^2 + x - 3 \)[/tex].
1. Start by substituting [tex]\( 6t \)[/tex] for [tex]\( x \)[/tex] in the function.
[tex]\[
f(6t) = 3(6t)^2 + (6t) - 3
\][/tex]
2. Compute [tex]\( (6t)^2 \)[/tex].
[tex]\[
(6t)^2 = 36t^2
\][/tex]
3. Multiply by 3.
[tex]\[
3 \cdot 36t^2 = 108t^2
\][/tex]
4. Now, substitute these values back into the function.
[tex]\[
f(6t) = 108t^2 + 6t - 3
\][/tex]
For [tex]\( t = 1 \)[/tex]:
[tex]\[
f(6t) = 108 \cdot 1^2 + 6 \cdot 1 - 3
\][/tex]
[tex]\[
f(6t) = 108 + 6 - 3
\][/tex]
[tex]\[
f(6t) = 111
\][/tex]
Thus, [tex]\( f(6t) = 111 \)[/tex].
