Questions: 17. Which of the following is a polynomial function? a. f(x)=x^2+3x-7 b. f(x)=3x^3-2x^-2+x c. f(x)=-5x^-4-4^x d. f(x)=2x^2-4^x

Solution Steps

A polynomial function is a function of the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \), where:

• The exponents \( n, n-1, \dots, 1, 0 \) are non-negative integers.
• The coefficients \( a_n, a_{n-1}, \dots, a_1, a_0 \) are real numbers.
• The variable \( x \) does not appear in the denominator, as an exponent, or inside a radical.

\( f(x) = x^{2} + 3x - 7 \):

• All exponents (2, 1, and 0) are non-negative integers.
• The function satisfies the definition of a polynomial.

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

• The term \( x^{-2} \) has a negative exponent, which violates the definition of a polynomial.
• Therefore, this is not a polynomial function.

\( f(x) = -5x^{-4} - 4^{x} \):

• The term \( x^{-4} \) has a negative exponent, which violates the definition of a polynomial.
• The term \( 4^{x} \) has the variable \( x \) as an exponent, which also violates the definition.
• Therefore, this is not a polynomial function.

Final Answer