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Which are cubic functions? Check all that apply.

A. [tex]f(x) = -4 - 2x + 5x^3[/tex]

B. [tex]f(x) = -3^x - 1[/tex]

C. [tex]f(x) = -x^2 + 5x - x^3[/tex]

D. [tex]f(x) = x^2 - 6x^3 + 2x^4 + 1[/tex]

E. [tex]f(x) = 2x^3 + 3x^2 - x - 3[/tex]

F. [tex]f(x) = 3x^2 - 1 - 8x[/tex]

Sagot :

GinnyAnsweridn
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]
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