Question

if $f(x) = x^2 + 3$, find and simplify the following: (a) $f(t + 8) =$ (b) $f(t^3 + 8) =$ (c) $f(3) =$ (d) $3f(t) =$ (e) $(f(t))^2 + 8 =$

Explanation:

Step1: Substitute $x=t+8$ into $f(x)$

$f(t+8)=(t+8)^2+3$

Step2: Expand and simplify

$f(t+8)=t^2+16t+64+3=t^2+16t+67$

Step1: Substitute $x=t^3+8$ into $f(x)$

$f(t^3+8)=(t^3+8)^2+3$

Step2: Expand and simplify

$f(t^3+8)=t^6+16t^3+64+3=t^6+16t^3+67$

Step1: Substitute $x=3$ into $f(x)$

$f(3)=3^2+3$

Step2: Calculate and simplify

$f(3)=9+3=12$

Step1: Substitute $x=t$ into $f(x)$, multiply by 3

$3f(t)=3(t^2+3)$

Step2: Distribute and simplify

$3f(t)=3t^2+9$

Step1: Substitute $x=t$ into $f(x)$, square it

$(f(t))^2=(t^2+3)^2$

Step2: Expand, add 8, simplify

$(f(t))^2+8=t^4+6t^2+9+8=t^4+6t^2+17$

Answer:

(a) $t^2+16t+67$
(b) $t^6+16t^3+67$
(c) $12$
(d) $3t^2+9$
(e) $t^4+6t^2+17$