Question

which function rule matches the table?
$f(x)=3 - x^3$
$f(x)=9 - 5x$
$f(x)=3 - x^2$
$f(x)=-\frac{1}{2}x$
$f(x)=x^2 - 5$

$x$	$f(x)$
3	-6
4	-13
5	-22

Explanation:

Step1: Test \( x = 2 \) in each function

• For \( f(x)=3 - x^{3} \): \( 3 - 2^{3}=3 - 8=-5

eq - 1 \)

• For \( f(x)=9 - 5x \): \( 9-5\times2 = 9 - 10=-1 \)
• For \( f(x)=3 - x^{2} \): \( 3-2^{2}=3 - 4=-1\) (so far, but check next \( x \))
• For \( f(x)=-\frac{1}{2}x \): \( -\frac{1}{2}\times2=-1\) (so far, check next \( x \))
• For \( f(x)=x^{2}-5 \): \( 2^{2}-5 = 4 - 5=-1\) (so far, check next \( x \))

Step2: Test \( x = 3 \) in remaining functions

• For \( f(x)=3 - x^{2} \): \( 3-3^{2}=3 - 9=-6\) (so far, check next \( x \))
• For \( f(x)=-\frac{1}{2}x \): \( -\frac{1}{2}\times3=-\frac{3}{2}

eq - 6 \)

• For \( f(x)=x^{2}-5 \): \( 3^{2}-5 = 9 - 5 = 4

eq - 6 \)

• For \( f(x)=9 - 5x \): \( 9-5\times3=9 - 15=-6 \)

Step3: Test \( x = 4 \) in remaining functions

• For \( f(x)=3 - x^{2} \): \( 3-4^{2}=3 - 16=-13 \)
• For \( f(x)=9 - 5x \): \( 9-5\times4=9 - 20=-11

eq - 13 \)

Step4: Test \( x = 5 \) in \( f(x)=3 - x^{2} \)

\( 3-5^{2}=3 - 25=-22 \), which matches the table. Wait, but earlier when \( x = 3 \), \( f(x)=3 - x^{2}=-6 \) and \( f(x)=9 - 5x=-6 \). Wait, let's re - check \( x = 4 \) for \( f(x)=9 - 5x \): \( 9-5\times4=9 - 20=-11\), but the table has \( f(4)=-13 \). For \( f(x)=3 - x^{2} \), when \( x = 4 \), \( 3 - 16=-13 \) (matches), \( x = 5 \): \( 3 - 25=-22 \) (matches). Wait, but when \( x = 2 \), \( f(x)=3 - 2^{2}=-1 \) (matches), \( x = 3 \): \( 3 - 9=-6 \) (matches), \( x = 4 \): \( 3 - 16=-13 \) (matches), \( x = 5 \): \( 3 - 25=-22 \) (matches). Wait, but let's re - check the first function: \( f(x)=3 - x^{3} \) at \( x = 2 \) is \( 3 - 8=-5
eq - 1 \). The function \( f(x)=9 - 5x \) at \( x = 4 \) is \( 9-20=-11
eq - 13 \). The function \( f(x)=3 - x^{2} \) works for all \( x \) values in the table. Wait, but let's check again:

For \( x = 2 \): \( 3-2^{2}=3 - 4=-1 \) (matches)

For \( x = 3 \): \( 3-3^{2}=3 - 9=-6 \) (matches)

For \( x = 4 \): \( 3-4^{2}=3 - 16=-13 \) (matches)

For \( x = 5 \): \( 3-5^{2}=3 - 25=-22 \) (matches)

Wait, but initially, I thought maybe I made a mistake. Let's confirm each function:

1. \( f(x)=3 - x^{3} \):

• \( x = 2 \): \( 3-8=-5

eq - 1 \) → Eliminate.

1. \( f(x)=9 - 5x \):

• \( x = 2 \): \( 9 - 10=-1 \)
• \( x = 3 \): \( 9 - 15=-6 \)
• \( x = 4 \): \( 9 - 20=-11

eq - 13 \) → Eliminate.

1. \( f(x)=3 - x^{2} \):

• \( x = 2 \): \( 3 - 4=-1 \)
• \( x = 3 \): \( 3 - 9=-6 \)
• \( x = 4 \): \( 3 - 16=-13 \)
• \( x = 5 \): \( 3 - 25=-22 \) → All match.

1. \( f(x)=-\frac{1}{2}x \):

• \( x = 2 \): \( - 1 \) (matches), \( x = 3 \): \( - 1.5

eq - 6 \) → Eliminate.

1. \( f(x)=x^{2}-5 \):

• \( x = 2 \): \( 4 - 5=-1 \) (matches), \( x = 3 \): \( 9 - 5 = 4

eq - 6 \) → Eliminate.

Answer:

\( f(x)=3 - x^{2} \)