To determine which of the given functions are cubic functions, we need to recall that a cubic function is a polynomial of degree 3. This means it is of the form [tex]\( ax^3 + bx^2 + cx + d \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] are constants, and the highest power of [tex]\( x \)[/tex] is [tex]\( 3 \)[/tex].
Let’s analyze each function step by step:
1. [tex]\( f(x) = -4 - 2x + 5x^3 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^3 \)[/tex], and there are no higher powers of [tex]\( x \)[/tex].
- Therefore, this function is a cubic function.
2. [tex]\( f(x) = -3^x - 1 \)[/tex]
- The term [tex]\( 3^x \)[/tex] is an exponential term, not a polynomial term, as the variable [tex]\( x \)[/tex] is in the exponent.
- Since it is not a polynomial, it cannot be a cubic function.
3. [tex]\( f(x) = -x^2 + 5x - x^3 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^3 \)[/tex], and there are no higher powers of [tex]\( x \)[/tex].
- Therefore, this function is a cubic function.
4. [tex]\( f(x) = x^2 - 6x^3 + 2x^4 + 1 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^4 \)[/tex].
- Since the highest power of [tex]\( x \)[/tex] is [tex]\( 4 \)[/tex], this function is a quartic function, not a cubic function.
5. [tex]\( f(x) = 2x^3 + 3x^2 - x - 3 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^3 \)[/tex], and there are no higher powers of [tex]\( x \)[/tex].
- Therefore, this function is a cubic function.
6. [tex]\( f(x) = 3x^2 - 1 - 8x \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex].
- Since the highest power of [tex]\( x \)[/tex] is [tex]\( 2 \)[/tex], this function is a quadratic function, not a cubic function.
From this analysis, the cubic functions among the given options are:
[tex]\[ f(x) = -4 - 2x + 5x^3 \][/tex]
[tex]\[ f(x) = -x^2 + 5x - x^3 \][/tex]
[tex]\[ f(x) = 2x^3 + 3x^2 - x - 3 \][/tex]
Therefore, the cubic functions are:
[tex]\[ f(x) = -4 - 2x + 5x^3 \][/tex]
[tex]\[ f(x) = -x^2 + 5x - x^3 \][/tex]
[tex]\[ f(x) = 2x^3 + 3x^2 - x - 3 \][/tex]
