Problem 26 - Calculus

Question

In Exercises \(24-26,\) choose the functions that are in the given family, assuming \(a, b,\) and \(c\) are constants. $$\begin{array}{ll}f(x)=\sqrt{x^{4}+16} & g(x)=a x^{23} \\\h(x)=-\frac{1}{5^{x-2}} & p(x)=\frac{a^{3} b^{x}}{c} \\\q(x)=\frac{a b^{2}}{c} & r(x)=-x+b-\sqrt{c x^{4}}\end{array}$$ Linear

Step-by-Step Solution

Answer

Function \( q(x)=\frac{a b^{2}}{c} \) is linear because it is a constant.

Step 1: Understanding Linear Functions

A linear function is in the form \( f(x) = mx + b \) where \( m \) and \( b \) are constants. It represents a straight line when graphed.

Step 2: Identify the linear functions

Examine each function to see if it fits the linear equation form:- \( f(x)=\sqrt{x^{4}+16} \) is not linear because it involves a square root of a power of \( x \).- \( g(x)=a x^{23} \) is not linear because the degree of \( x \) is 23.- \( h(x)=-\frac{1}{5^{x-2}} \) is not linear as it involves an exponential expression.- \( p(x)=\frac{a^{3} b^{x}}{c} \) is not linear due to the presence of \( b^x \).- \( q(x)=\frac{a b^{2}}{c} \) is a constant function because there is no \( x \)-variable.- \( r(x)=-x+b-\sqrt{c x^{4}} \) involves \( \sqrt{c x^{4}} \), making it nonlinear.

Step 3: Determine the constant or linear function

A constant function, such as \( q(x)=\frac{a b^{2}}{c} \), is a special type of linear function since it can be expressed as \( f(x) = 0x + b \), where \( b = \frac{a b^{2}}{c} \) is constant and does not depend on \( x \).

Key Concepts

Linear Equation Form

Linear functions are a fundamental concept in mathematics. The standard form for a linear equation is given by \( f(x) = mx + b \). Here, \( m \) represents the slope of the line, while \( b \) is the y-intercept, which is the point where the line crosses the y-axis. Linear functions produce straight lines when plotted on a graph.

To identify a linear function, you must ensure that the only power of \( x \) permissible is 1. This means no square roots, exponents other than 1, or transformations of \( x \) that would result in any non-linear characteristics.

• Example 1: \( f(x) = 2x + 3 \) is a linear function.
• Example 2: \( f(x) = 5 - x, \) another linear form, shows a negative slope.

In essence, linear functions highlight direct proportionality between \( x \) and \( f(x) \).

Constant Functions

Constant functions are a unique subset of linear functions. They appear in the form \( f(x) = b \) where \( m = 0 \).

This form indicates that the function’s output remains constant, regardless of the input value for \( x \). Graphically, constant functions are represented by horizontal lines, revealing no slope.

• Example: \( f(x) = 4 \) represents a horizontal line at \( y = 4 \).
• This showcases that for any value of \( x, \) \( f(x) \) remains 4.

Hence, although constant functions have a linear equation form, the lack of an \( x \) variable signifies that their graph is a flat line. This simplicity makes them easy to recognize and utilize.

Exponential Expressions

Exponential expressions are mathematical statements where the variable resides in the exponent. They are not linear and do not conform to the linear equation form \( mx + b \). Instead, these functions grow by consistent multiplicative factors and demonstrate rapid increase or decrease.

One can identify exponential expressions by their standard form: \( f(x) = a \, b^x \), where \( a \) is a coefficient and \( b \) is the base raised to the power of \( x \).

• Example: \( f(x) = 2^x \) represents exponential growth as \( x \) increases.
• Conversely, \( g(x) = \frac{1}{2^x} \) exemplifies exponential decay.

These functions are crucial in modeling situations where growth accelerates, such as in population dynamics or compound interest scenarios.