Sure, let’s determine the value of [tex]\( f(t+1) \)[/tex] for the function [tex]\( f(x) = 7x^2 - 6 \)[/tex].
First, we need to substitute [tex]\( x \)[/tex] with [tex]\( t+1 \)[/tex] in the given function. This involves replacing [tex]\( x \)[/tex] in the expression [tex]\( 7x^2 - 6 \)[/tex] with [tex]\( t+1 \)[/tex].
Starting with the function:
[tex]\[
f(x) = 7x^2 - 6
\][/tex]
Substituting [tex]\( x = t+1 \)[/tex]:
[tex]\[
f(t+1) = 7(t+1)^2 - 6
\][/tex]
Next, we need to expand [tex]\( (t+1)^2 \)[/tex].
So, we have:
[tex]\[
(t+1)^2 = t^2 + 2t + 1
\][/tex]
Substitute back:
[tex]\[
f(t+1) = 7(t^2 + 2t + 1) - 6
\][/tex]
Distribute the 7:
[tex]\[
7(t^2 + 2t + 1) = 7t^2 + 14t + 7
\][/tex]
Now, subtract 6:
[tex]\[
f(t+1) = 7t^2 + 14t + 7 - 6
\][/tex]
Simplify the constant terms:
[tex]\[
f(t+1) = 7t^2 + 14t + 1
\][/tex]
Thus, the value of [tex]\( f(t+1) \)[/tex] is:
[tex]\[
f(t+1) = 7t^2 + 14t + 1
\][/tex]
