To determine whether each given function is a polynomial function, we need to check if the function can be written as the sum of terms of the form [tex]\( ax^n \)[/tex] where [tex]\( a \)[/tex] is a real coefficient and [tex]\( n \)[/tex] is a non-negative integer.
Let's analyze each function step-by-step:
### (a) [tex]\( f(x) = x^9 - 7 + 3x^{-6} \)[/tex]
- The term [tex]\( x^9 \)[/tex] is fine since it is of the form [tex]\( ax^n \)[/tex] with [tex]\( a=1 \)[/tex] and [tex]\( n=9 \)[/tex], which is a non-negative integer.
- The constant term [tex]\(-7\)[/tex] is also acceptable in a polynomial function.
- The term [tex]\( 3x^{-6} \)[/tex] involves a negative exponent ([tex]\( -6 \)[/tex]), which is not allowed in polynomials.
Based on the analysis, since there is a term with a negative exponent, the function [tex]\( f(x) = x^9 - 7 + 3x^{-6} \)[/tex] is not a polynomial.
### (b) [tex]\( v(x) = 3 \sqrt{x} - 6x^5 \)[/tex]
- The term [tex]\( 3 \sqrt{x} \)[/tex] can be rewritten as [tex]\( 3x^{1/2} \)[/tex]. The exponent [tex]\( 1/2 \)[/tex] is a non-integer, which is not allowed in polynomials.
- The term [tex]\( -6x^5 \)[/tex] is of the form [tex]\( ax^n \)[/tex] with [tex]\( a = -6 \)[/tex] and [tex]\( n = 5 \)[/tex], which is a non-negative integer.
Since the term [tex]\( 3 \sqrt{x} \)[/tex] involves a non-integer exponent, the function [tex]\( v(x) = 3 \sqrt{x} - 6x^5 \)[/tex] is not a polynomial.
### (c) [tex]\( h(x) = -\frac{6}{x^3} \)[/tex]
- The term [tex]\( -\frac{6}{x^3} \)[/tex] can be rewritten as [tex]\( -6x^{-3} \)[/tex]. The exponent [tex]\( -3 \)[/tex] is a negative integer, which is not allowed in polynomials.
Therefore, the function [tex]\( h(x) = -\frac{6}{x^3} \)[/tex] is not a polynomial.
### (d) [tex]\( u(x) = 7x^6 \)[/tex]
- The term [tex]\( 7x^6 \)[/tex] is of the form [tex]\( ax^n \)[/tex] with [tex]\( a = 7 \)[/tex] and [tex]\( n = 6 \)[/tex], which is a non-negative integer.
Since all terms in the function meet the polynomial criteria, the function [tex]\( u(x) = 7x^6 \)[/tex] is a polynomial.
### Summary:
\begin{tabular}{|l|c|}
\hline \textbf{Function} & \textbf{Is the function a polynomial?} \\
\hline (a) [tex]\( f(x) = x^9 - 7 + 3 x^{-6} \)[/tex] & No \\
\hline (b) [tex]\( v(x) = 3 \sqrt{x} - 6 x^5 \)[/tex] & No \\
\hline (c) [tex]\( h(x) = -\frac{6}{x^3} \)[/tex] & No \\
\hline (d) [tex]\( u(x) = 7 x^6 \)[/tex] & Yes \\
\hline
\end{tabular}
