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Chapter 2: Problem 72

Determine if the function is linear, constant, or neither. a. \(m(x)=5 x+1\) b. \(n(x)=\frac{5}{x}+1\) c. \(p(x)=5\) d. \(q(x)=5 x\)

Short Answer

Expert verified
a. Linear, b. Neither, c. Constant, d. Linear

Step by step solution

01

Identify Linearity in m(x)

Check if the function is of the form \(m(x) = ax + b\). Here, \(m(x) = 5x + 1\) is a linear function because it is in the form of \(ax + b\) where \(a = 5\) and \(b = 1\).
02

Identify Linearity in n(x)

Check if the function is of the form of a linear function \(c x + d\). Here, \(n(x) = \frac{5}{x} + 1\) is not linear because it involves the variable \(x\) in the denominator.
03

Identify Linearity in p(x)

Check if the function is constant. Here, \(p(x) = 5\) does not change with \(x\), thus it is a constant function.
04

Identify Linearity in q(x)

Check if the function is of the form \(d x\). Here, \(q(x) = 5x\) is a linear function because it matches the form \(ax\) where \(a = 5\) and \(b = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Functions
A constant function is one where the value does not change regardless of the input variable, usually denoted as \( f(x) = c \) where \( c \) is a constant. For example, \( p(x) = 5 \) is a constant function because whenever you input any value for \( x \), the output remains 5. This type of function is represented by a horizontal line on a graph, illustrating that the y-value remains consistent regardless of the x-value.
In the context of our exercise, \( p(x) = 5 \) is correctly identified as a constant function. It鈥檚 important to recognize these functions because they represent real-life scenarios where the output is unaffected by changes in the input.
Identifying Function Types
Identifying the type of function is crucial for solving mathematical problems and understanding their behavior. Functions can generally be linear, constant, or neither. Here's a concise way to differentiate them:
  • Linear Functions: A function of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. For example, \( m(x) = 5x + 1 \) and \( q(x) = 5x \) are linear because they fit the form \( ax + b \).
  • Constant Functions: A function where the output value remains the same, irrespective of the input. As mentioned, \( p(x) = 5 \) is a constant function.
  • Neither: Functions that do not fit into the linear or constant categories. In our example, \( n(x) = \frac{5}{x} + 1 \) does not fit the linear form because the variable \( x \) is in the denominator.
Knowing these types helps in determining their graphical representations and behavior under different operations.
Mathematical Forms
Understanding the different mathematical forms of functions aids in identifying patterns and solving equations more effectively. Let's break down a few:
  • Linear Form: Represented as \( ax + b \). If both \( a \) and \( b \) are constants, the function graphs as a straight line. For instance, \( m(x) = 5x + 1 \) shows a line with a slope of 5 and a y-intercept of 1.
  • Constant Form: Denoted as \( c \), where \( c \) is a constant. This is a special linear form where the slope is 0, meaning the graph is a horizontal line. For example, \( p(x) = 5 \).
  • Other Forms: More complex functions, like rational functions (e.g., \( n(x) = \frac{5}{x} + 1 \)), cannot be classified as linear or constant due to their inclusion of variables in numerators, denominators, or other non-linear configurations.
Breaking down functions into these forms simplifies the process of solving and understanding them, making it easier to predict their behavior across different input values.

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