Question

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

2. (a) \(f\left( t \right) = \frac{{{\bf{3}}{t^{\bf{2}}} + {\bf{2}}}}{t}\)

(b) \(h\left( r \right) = {\bf{2}}.{{\bf{3}}^r}\)

(c) \(s\left( t \right) = \sqrt {t + {\bf{4}}} \)

(d) \(y = {x^{\bf{4}}} + 5\)

(e) \(g\left( x \right) = \sqrt({\bf{3}}){x}\)

(f) \(y = \frac{{\bf{1}}}{{{x^{\bf{2}}}}}\)

Short answer

(a) The function is a rational function, also it is an algebraic function.

(b) The function is an exponential function.

(c) The function is an algebraic function.

(d) The function is a polynomial function of degree 4.

(e) The function is a power function, also it is an algebraic function.

(f) The function is a rational function, also it is an power function.

Step-by-step solution

Classify the function (a)

The given function is \(f\left( t \right) = \frac{{3{t^2} + 2}}{t}\).

It is observed that the function \(f\left( t \right) = \frac{{3{t^2} + 2}}{t}\) consists of algebraic operations. So, this function is an algebraic function consists of polynomial of degree 2 in the numerator, and polynomial of degree 1 in the denominator.

Also, this function is of the form\(f\left( t \right) = \frac{{P\left( t \right)}}{{Q\left( t \right)}}\), where\(P\left( t \right) = 3{t^2} + 2\), and\(Q\left( t \right) = t\). So, the function is a rational function.

Thus, the given function is an algebraic function, also arational function.

Classify the function (b)

The given function is \(h\left( r \right) = {2.3^r}\).

It is observed that the function\(h\left( r \right) = {2.3^r}\)is of the exponential function form\(f\left( x \right) = {b^x}\), where\(b\)is a positive constant.

Thus, the given function is an exponential function.

Classify the function (c)

It is observed that the function \(s\left( t \right) = \sqrt {t + 4} \) consists of algebraic operations. So, this function is an algebraic function consists of root of a polynomial.

Thus, the given function is an algebraic function.

Classify the function (d)

The given function \(y = f\left( x \right) = {x^4} + 5\) is of the polynomial form \(f\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ... + {a_0}\). So, \(n = 4\).

Thus, the given function is a polynomial function of degree 4.

Also, the given function consistsof algebraic operations. So, this function is also an algebraic function.

Therefore, the function \(f\left( x \right) = {x^4} + 5\) is a polynomial function of degree 4, it is also an algebraic function.

Classify the function (e)

It is observed that the function \(g\left( x \right) = \sqrt(3){x}\) consists of algebraic operations. So, this function is an algebraic function consists of root of a polynomial.

Also, the given function can be classified as a root function, and power function because it can be written as\(g\left( x \right) = \sqrt(3){x} = {x^{1/3}}\).

Thus, the given function is an algebraic function, also power and root function

Classify the function (f)

The given function is \(y = \frac{1}{{{x^2}}}\).

The function is of the form\(f\left( x \right) = \frac{{P\left( x \right)}}{{Q\left( x \right)}}\), where\(P\left( x \right) = 1\), and\(Q\left( x \right) = {x^2}\). So, the function is a rational function.

Also, the function can be written as\(y = \frac{1}{{{x^2}}} = {x^{ - 2}}\). So, the function is a power function.

Therefore, the function is a rational function, also it is an power function.