Problem 6 - Intermediate Algebra

Question

Which of the following functions are exponential functions? a. \(f(x)=x^{2}\) b. \(g(x)=4 x\) c. \(h(x)=8^{x}\) d. \(s(x)=\frac{1}{x}\) e. \(T(x)=(0.92)^{x+1}\) t. \(r(x)=x^{3}\) g. \(P(x)=\sqrt{x}\) h. \(d(x)=|x|\)

Step-by-Step Solution

Answer

Functions c (\(h(x)=8^{x}\)) and e (\(T(x)=(0.92)^{x+1}\)) are exponential.

Step 1: Define an Exponential Function

An exponential function is a mathematical function in the form \(f(x) = a \cdot b^x\), where \(a\) is a constant, \(b\) is a positive real number (\(b>0\)) and \(beq 1\), and \(x\) is the exponent.

Step 2: Analyze Each Function

Let's go through each function to determine if it fits the exponential form:- **a.** \(f(x)=x^{2}\) is a polynomial, not exponential.- **b.** \(g(x)=4x\) is linear, not exponential.- **c.** \(h(x)=8^{x}\) fits the exponential form with \(a=1\) and \(b=8\).- **d.** \(s(x)=\frac{1}{x}\) is a rational function, not exponential.- **e.** \(T(x)=(0.92)^{x+1}\) fits the exponential form with \(a=1\) and \(b=0.92\), though it has an adjustment with \(x+1\).- **t.** \(r(x)=x^{3}\) is a polynomial, not exponential.- **g.** \(P(x)=\sqrt{x}\) is a radical function, not exponential.- **h.** \(d(x)=|x|\) represents the absolute value function, not exponential.

Step 3: Identify Exponential Functions

From our analysis:- **c.** \(h(x)=8^{x}\) is an exponential function.- **e.** \(T(x)=(0.92)^{x+1}\) is also an exponential function (note the adjustment \(x+1\), which still maintains exponential characteristics).

Key Concepts

Polynomial Functions

Polynomial functions are essential building blocks in mathematics, representing equations composed of variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents. They are expressed as a sum of terms, each consisting of a constant coefficient and a variable raised to a whole number power.
A polynomial function takes the form:

• General form: \[ p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \]
• Example: \[ f(x) = x^2 \, \text{or}\, f(x) = x^3 \]

Here, each "a" is a constant, known as a coefficient, and "x" is the variable.
These equations can model various natural and man-made phenomena. However, exponential growth or decay cannot be modelled with polynomial functions alone. Understanding polynomials helps in recognizing their distinct structure from other functions such as exponential ones.

Linear Functions

Linear functions are the simplest form of polynomial functions, as they involve only the first power of the variable. These functions are characterized by a constant rate of change, resulting in a straight line when graphed. They can be represented in the standard form:

• General form: \[ f(x) = mx + b \]
• Where "m" represents the slope, and "b" is the y-intercept.
• Example: \[ g(x) = 4x \]

In linear functions, the relationship between variables is directly proportional. This means that doubling the input will double the output, maintaining a consistent ratio. Linear functions are essential in real-world applications, especially for modeling situations with constant change. They should not be confused with exponential functions, which involve variables in the exponent.

Rational Functions

Rational functions are intriguing because they consist of ratios of polynomial functions. They are expressed as the quotient of two polynomials, where the denominator is not zero, giving rise to unique characteristics, such as asymptotes and discontinuities.

• General form: \[ R(x) = \frac{p(x)}{q(x)} \]
• Example: \[ s(x) = \frac{1}{x} \]

These functions can model situations involving rates or proportions and have more complex behaviors compared to polynomial and linear functions. Vertical asymptotes occur where the denominator equals zero, giving insight into different limits and extreme values of the function. However, they still vastly differ from exponential functions, where the variable itself is found in the exponent of the expression.

Radical Functions

Radical functions involve roots of variables, primarily focusing on square roots, cube roots, and so forth. They are expressed using radical symbols or fractional exponents.

• General form: \[ P(x) = \sqrt[n]{x} \, \text{or}\, x^{1/n} \]
• Example: \[ P(x) = \sqrt{x} \]

These functions can describe various natural phenomena, such as wave patterns and growth relationships that are not strictly linear or polynomial. Radical functions introduce complexities such as restricted domains, especially when dealing with square roots of non-positive numbers, since they cannot yield real numbers. Despite sharing some properties, they remain distinct from exponential functions, which focus on continuous multiplicative growth.