Question

which of the following is an example of a polynomial function?
$f(x) = 5x^3 + 2x^2 - 6x + 7$
$g(x) = \frac{x^2 + 2x - 3}{x - 4}$
$h(x) = 750(1.04)^x$
$p(x) = \log_3(x)^2$

Explanation:

Step1: Recall the definition of a polynomial function

A polynomial function is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non - negative integer exponents of variables. In general, a polynomial function in one variable \(x\) can be written as \(f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\), where \(n\) is a non - negative integer and \(a_i\) are constants.

Step2: Analyze \(f(x) = 5x^{3}+2x^{2}-6x + 7\)

For the function \(f(x)=5x^{3}+2x^{2}-6x + 7\), we have terms with non - negative integer exponents of \(x\) (the exponents are \(3\), \(2\), \(1\), and \(0\) for the constant term \(7 = 7x^{0}\)) and the operations are addition, subtraction, and multiplication of coefficients and powers of \(x\). So this function follows the form of a polynomial function.

Step3: Analyze \(g(x)=\frac{x^{2}+2x - 3}{x - 4}\)

The function \(g(x)\) is a rational function (a ratio of two polynomials). Since it has a variable in the denominator (\(x-4\)), it does not satisfy the definition of a polynomial function (polynomial functions do not have variables in the denominator).

Step4: Analyze \(h(x)=750(1.04)^{x}\)

The function \(h(x)\) is an exponential function. In an exponential function, the variable \(x\) is in the exponent, which is not allowed in a polynomial function (in polynomial functions, the variable is in the base and the exponent is a non - negative integer).

Step5: Analyze \(p(x)=\log_{3}(x)^{2}\)

The function \(p(x)\) is a logarithmic function. Logarithmic functions involve the logarithm of a variable, which is not part of the definition of a polynomial function (polynomial functions are made up of powers of the variable, addition, subtraction, and multiplication by constants).

Answer:

\(f(x)=5x^{3}+2x^{2}-6x + 7\) (the option corresponding to this function)