Question

analyzing functions using a table
the function ( f(x) = x^5 + (x + 3)^2 ) is used to create this table.

( x )	( -2 )	( -1 )	( 0 )	( 1 )
( f(x) )	( -31 )	( ? )	( 9 )	( 17 )

which value completes the table?
( circ -17 )
( circ -3 )
( circ 1 )
( circ 3 )

Explanation:

Step1: Identify the function and x-value

We have the function \( f(x) = x^5 + (x + 3)^2 \) and we need to find \( f(-1) \) (since the x - value for the unknown \( f(x) \) is \( x=-1 \)).

Step2: Substitute \( x = - 1 \) into the function

First, calculate \( x^5 \) when \( x=-1 \): \( (-1)^5=-1 \) (because any odd power of - 1 is - 1).
Then, calculate \( (x + 3)^2 \) when \( x=-1 \): \( (-1 + 3)^2=(2)^2 = 4 \) (using the formula \( (a + b)^2=a^{2}+2ab + b^{2} \), here \( a=-1 \), \( b = 3 \), so \( (-1+3)^2=2^2 = 4 \)).

Step3: Add the two results

Now, add the two parts together: \( f(-1)=(-1)+4 = 3 \)? Wait, no, wait. Wait, \( (-1)^5=-1 \), \( ( - 1+3)^2=(2)^2 = 4 \), so \( f(-1)=-1 + 4=3 \)? Wait, but let's re - check. Wait, \( x=-1 \), so \( x^5=(-1)^5=-1 \), \( (x + 3)^2=(-1 + 3)^2=2^2 = 4 \), so \( f(-1)=-1+4 = 3 \)? Wait, but let's check the function again. Wait, the function is \( f(x)=x^5+(x + 3)^2 \). So substituting \( x=-1 \):

\( f(-1)=(-1)^5+(-1 + 3)^2=-1+(2)^2=-1 + 4 = 3 \).

Wait, but let's check with other values to make sure the function is correct. Let's check \( x = 0 \): \( f(0)=0^5+(0 + 3)^2=0 + 9 = 9 \), which matches the table. \( x = 1 \): \( f(1)=1^5+(1 + 3)^2=1+16 = 17 \), which matches the table. \( x=-2 \): \( f(-2)=(-2)^5+(-2 + 3)^2=-32+1=-31 \), which matches the table. So the function is correct. Then for \( x=-1 \), \( f(-1)=(-1)^5+(-1 + 3)^2=-1 + 4 = 3 \).

Answer:

3