Question

evaluate the function $f(x)=x^2 + 3x + 1$ at the given values of the independent variable and simplify.

a. $f(-3)$

b. $f(x + 1)$

c. $f(-x)$

a. $f(-3) = \boxed{1}$ (simplify your answer)

b. $f(x + 1) = \square$ (simplify your answer)

Explanation:

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

$f(-3) = (-3)^2 + 3(-3) + 1$

Step2: Calculate each term

$(-3)^2=9$, $3(-3)=-9$, so $9 - 9 + 1$

Step3: Substitute $x+1$ into $f(x)$

$f(x+1) = (x+1)^2 + 3(x+1) + 1$

Step4: Expand and simplify the expression

$(x^2+2x+1) + (3x+3) + 1 = x^2+2x+1+3x+3+1 = x^2+5x+5$

Step5: Substitute $-x$ into $f(x)$

$f(-x) = (-x)^2 + 3(-x) + 1$

Step6: Simplify the expression

$(-x)^2=x^2$, $3(-x)=-3x$, so $x^2-3x+1$

Answer:

a. $f(-3)=1$
b. $f(x+1)=x^2+5x+5$
c. $f(-x)=x^2-3x+1$