Question

think about the function $f(x) = 10 - x^3$.
what is the input, or independent variable?
\\(\circ\\) $f(x)$
\\(\circ\\) $x$
\\(\circ\\) $y$
done
what is the output, or dependent variable or quantity?
\\(\circ\\) $x$
\\(\circ\\) $f(x)$
\\(\circ\\) $y$
done
what does the notation $f(2)$ mean?
\\(\circ\\) multiply $f$ by 2
\\(\circ\\) the output ($y$-value) when $x = 2$
\\(\circ\\) the value of $x$ when the output is 2
done
evaluate $f(2) = \square$.
done

Explanation:

For the question "Evaluate \( f(2) \)":

Step 1: Substitute \( x = 2 \) into the function \( f(x)=10 - x^{3} \)

We replace \( x \) with 2 in the function. So we get \( f(2)=10-(2)^{3} \).

Step 2: Calculate \( 2^{3} \)

We know that \( 2^{3}=2\times2\times2 = 8 \). So now the expression becomes \( f(2)=10 - 8 \).

Step 3: Subtract

\( 10-8 = 2 \).

In a function \( y = f(x) \) (or \( f(x) \) representing the output), the independent variable is the one that we can choose freely, and it is the input to the function. In the function \( f(x)=10 - x^{3} \), the variable \( x \) is the input variable. So the correct option is "x".

The notation \( f(a) \) in a function \( f(x) \) means the output (the \( y \) - value) of the function when the input \( x=a \). So when \( a = 2 \), \( f(2) \) means the output ( \( y \) - value) when \( x = 2 \). So the correct option is "the output (y - value) when \( x = 2 \)".

Answer:

\( 2 \)

For the multiple - choice questions:

1. "What is the input, or independent variable?"