Question

(e) $y = \frac{\sqrt{x}}{x^2 + 8}$ \bigcirc power function \bigcirc root function \bigcirc polynomial function of degree 2 \bigcirc polynomial function of degree 8 \bigcirc rational function \bigcirc algebraic function (f) $g(u) = \log_{10}(u)$ \bigcirc power function \bigcirc root function \bigcirc polynomial function of degree 10 \bigcirc algebraic function \bigcirc exponential function \bigcirc logarithmic function

Explanation:

Step1: Analyze function (e)

Rewrite $y=\frac{\sqrt{x}}{x^2+8}$ as $y=\frac{x^{\frac{1}{2}}}{x^2+8}$. An algebraic function involves roots, polynomials, and their ratios. This function is a ratio of a root function (a type of algebraic function) and a polynomial, so it falls under algebraic functions.

Step2: Analyze function (f)

The function $g(u)=\log_{10}(u)$ follows the form $\log_b(x)$, which is the definition of a logarithmic function.

Answer:

(e) algebraic function
(f) logarithmic function