Problem 6 - Elementary and 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. \(T(x)=(0.92)^{x+1}\) f. \(r(x)=x^{3}\) g. \(P(x)=\sqrt{x}\) h. \(d(x)=|x|\)

Step-by-Step Solution

Answer

The exponential functions are \( h(x) = 8^x \) and \( T(x) = (0.92)^{x+1} \).

Step 1: Understanding Exponential Functions

An exponential function is defined as any mathematical function of the form \( f(x) = a^x \), where \( a \) is a positive real number not equal to 1, and \( x \) is any real number. Therefore, we are looking for functions that can be rewritten in the form \( a^x \).

Step 1: Analyze Function f(x)

The function \( f(x) = x^2 \) is a polynomial function, not an exponential function, because the variable \( x \) is the base, not the exponent.

Step 2: Analyze Function g(x)

The function \( g(x) = 4x \) is a linear function because it can be written in the form \( g(x) = mx + b \). In this case, \( m = 4 \) and \( b = 0 \). This is not an exponential function.

Step 3: Analyze Function h(x)

The function \( h(x) = 8^x \) is in the form \( a^x \), where \( a = 8 \). Therefore, \( h(x) \) is an exponential function.

Step 4: Analyze Function s(x)

The function \( s(x) = \frac{1}{x} \) is a rational function, not an exponential function, as the variable \( x \) is in the denominator.

Step 5: Analyze Function T(x)

The function \( T(x) = (0.92)^{x+1} \) can be rewritten as \( (0.92)^x \times (0.92)^1 \), which is still an exponential function of the form \( a^x \). Hence, \( T(x) \) is an exponential function.

Step 6: Analyze Function r(x)

The function \( r(x) = x^3 \) is a polynomial function, not an exponential function, because the variable \( x \) is the base.

Step 7: Analyze Function P(x)

The function \( P(x) = \sqrt{x} \) can be rewritten as \( x^{1/2} \), which is a root function, not an exponential function.

Step 8: Analyze Function d(x)

The function \( d(x) = |x| \) represents the absolute value of \( x \). It is not an exponential function.

Key Concepts

Polynomial Functions

Polynomial functions are fundamental in algebra and calculus. They are expressions that consist of variables raised to whole number powers, multiplied by coefficients. The general form of a polynomial is:

• \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]
• "n" is a non-negative integer, indicating the degree.
• "a_n, a_{n-1}, ... , a_0" are constants.
• The degree dictates the graph's shape and the number of roots.

For example, the function \( r(x) = x^3 \) is a polynomial of degree 3, which signifies a cubic function. These functions can describe anything from the parabola of a quadratic equation to more complex shapes for higher degrees. They are not exponential because the variable is the base, not the exponent.

Linear Functions

Linear functions are the simplest form of polynomial functions. They graph as straight lines and are characterized by their constant rate of change or slope. The standard form of a linear function is:

• \[ g(x) = mx + b \]
• "m" represents the slope, which shows the rate of change.
• "b" is the y-intercept, the point where the line crosses the y-axis.

The function \( g(x) = 4x \), for instance, is a linear function where the slope \( m = 4 \) signifies that for every increase in \( x \), \( g(x) \) increases by 4.Linear functions are not exponential because they do not have a variable in the exponent, and their constant rate of change is a key differentiator.

Rational Functions

Rational functions are the quotient of two polynomial functions. This means they are expressed as:

• \[ f(x) = \frac{p(x)}{q(x)} \]
• "p(x)" and "q(x)" are polynomials, with "q(x)" not equaling zero.

These functions can have vertical asymptotes, points where the function is undefined, and horizontal asymptotes, indicating the function's behavior as \( x \) approaches infinity.The function \( s(x) = \frac{1}{x} \) is a simple rational function, illustrating how the variable in the denominator influences the function's behavior drastically, distinguishing it from exponential functions.

Root Functions

Root functions involve the extraction of a root and can also be seen as polynomial functions with fractional exponents. They are expressed in the form:

• \[ f(x) = x^{1/n} \]
• "n" represents the root degree, and the function represents the "n-th" root of \( x \).

For example, the function \( P(x) = \sqrt{x} \) is equivalent to \( x^{1/2} \). Root functions are generally used to model scenarios where diminishing returns or leveling off is observed, but they don't grow or decay exponentially since the exponent is a fraction rather than a variable.

Absolute Value Functions

Absolute value functions are unique in their representation of distance from zero, regardless of direction. They are defined mathematically as:

• \[ d(x) = |x| \]
• This means the function outputs \( x \) if \( x \geq 0 \), and \(-x \) if \( x < 0 \).

Graphically, this forms a "V" shape, making them particularly useful in representing real-world scenarios where only magnitudes are considered. They differ from exponential functions as they do not involve powers and solely focus on the unsigned value of numbers, emphasizing their nature of magnitude.