Question

table of x and y values
x | f(x)
-64 | -4
-27 | -3
-8 | -2
-1 | -1
0 | 0
1 | 1
8 | 2
27 | 3
64 | 4
125 | 5
given the table of values, determine the function.
(1 point)
○ $f(x) = x^3$
○ $f(x) = x^2$
○ $f(x) = \sqrt{x}$
○ $f(x) = \sqrt3{x}$

Explanation:

Step1: Analyze \( f(x)=x^3 \)

Check \( x = -4 \), \( (-4)^3=-64 \), but in table \( f(-64)=-4 \), not \( -64^3 \). So not this.

Step2: Analyze \( f(x)=x^2 \)

\( x=-64 \), \( (-64)^2=4096
eq -4 \). Also, square of negative is positive, but table has negative \( f(x) \) for negative \( x \). Eliminate.

Step3: Analyze \( f(x)=\sqrt{x} \)

Square root of negative \( x \) (like \( -64 \)) is not real, but table has \( f(-64)=-4 \). Eliminate.

Step4: Analyze \( f(x)=\sqrt[3]{x} \)

Check \( x=-64 \), \( \sqrt[3]{-64}=-4 \); \( x=-27 \), \( \sqrt[3]{-27}=-3 \); \( x=8 \), \( \sqrt[3]{8}=2 \); \( x=125 \), \( \sqrt[3]{125}=5 \). Matches all table values.

Answer:

\( \boldsymbol{f(x) = \sqrt[3]{x}} \) (the fourth option: \( f(x)=\sqrt[3]{x} \))