Suppose That The Function G Is Defined As Follows

Let's delve into the world of functions, specifically focusing on understanding and analyzing a function defined as 'g'. A well-defined function is a cornerstone of mathematics, enabling us to model relationships between different variables and make predictions based on observed patterns. In this exploration, we'll break down the components of a function definition, explore common representations, and work through examples to solidify our understanding.

Understanding Function Definition

At its core, a function is a rule that assigns a unique output value to each input value. This assignment must be unambiguous; for a given input, there can be only one output. The set of all possible input values is called the domain of the function, while the set of all possible output values is called the range.

When we say "suppose that the function g is defined as follows," we're essentially being provided with a specific rule that governs how the function g operates. This rule can be expressed in various forms:

• Equation: The most common way to define a function is using an equation. For example, g(x) = x^2 + 3x - 2. This equation explicitly tells us what to do with the input x to obtain the output g(x).

• Graph: A function can be represented graphically by plotting the input-output pairs as points on a coordinate plane. The vertical line test helps determine if a graph represents a function: if any vertical line intersects the graph more than once, it is not a function.

• Table: A table lists specific input values and their corresponding output values. This representation is useful when the function's rule is not easily expressible as an equation.

• Verbal Description: Sometimes, a function is defined using a verbal description of the rule. For instance, "g(x) is the absolute value of x."

• Piecewise Definition: Functions can be defined differently over different intervals of their domain. These are called piecewise functions. For example:

  g(x) = {
      x + 1, if x < 0
      x^2,   if x >= 0
  }
  

The notation g(x) signifies that g is a function, and x is the input variable. The output of the function when the input is x is denoted by g(x). Understanding this notation is crucial for working with functions.

Key Concepts & Terminology

Before diving into examples, let's clarify some related concepts:

• Independent Variable: The input variable (usually x) is called the independent variable. Its value is chosen freely from the domain.
• Dependent Variable: The output variable (usually g(x) or y) is called the dependent variable. Its value depends on the input value.
• Domain: The set of all possible input values for which the function is defined.
• Range: The set of all possible output values that the function can produce.
• Function Evaluation: Finding the output value of a function for a specific input value. This involves substituting the input value into the function's rule.
• Function Composition: Combining two functions by applying one function to the result of another. For example, if f(x) = x + 1 and g(x) = x^2, then the composition f(g(x)) = (x^2) + 1.
• Inverse Function: If a function g is one-to-one (meaning each output corresponds to a unique input), then it has an inverse function, denoted by g⁻¹(x), which "undoes" what g does. That is, g⁻¹(g(x)) = x and g(g⁻¹(x)) = x.

Working with Function Definitions: Examples

Let's illustrate these concepts with various examples of function definitions and how to work with them.

Example 1: Function Defined by an Equation

Suppose the function g is defined as follows:

g(x) = 3x - 5

• Function Evaluation: To find g(2), we substitute x = 2 into the equation:

  g(2) = 3(2) - 5 = 6 - 5 = 1
  

  So, g(2) = 1. Similarly, g(-1) = 3(-1) - 5 = -8.

• Finding the Domain: Unless otherwise specified, we assume the domain of g is all real numbers for which the expression 3x - 5 is defined. In this case, there are no restrictions on x (we can multiply any real number by 3 and subtract 5), so the domain is all real numbers, which can be written as (-∞, ∞).

• Finding the Range: Since the function is a linear function (a straight line), and the coefficient of x is not zero, the range is also all real numbers (-∞, ∞). Linear functions without restrictions on their domain will always have a range of all real numbers.

• Solving for x when g(x) is known: Suppose we want to find the value of x such that g(x) = 4. We set up the equation:

  3x - 5 = 4
  

  Adding 5 to both sides gives:

  3x = 9
  

  Dividing both sides by 3 gives:

  x = 3
  

  Therefore, g(3) = 4.

Example 2: Piecewise Function

Suppose the function g is defined as follows:

g(x) = {
    x^2,   if x < 1
    2x + 1, if x >= 1
}

• Function Evaluation: To find g(0), we note that 0 < 1, so we use the first part of the definition:

  g(0) = 0^2 = 0
  

  To find g(1), we note that 1 >= 1, so we use the second part of the definition:

  g(1) = 2(1) + 1 = 3
  

  To find g(2), we note that 2 >= 1, so we use the second part of the definition:

  g(2) = 2(2) + 1 = 5
  

• Finding the Domain: The function is defined for all real numbers because each real number either satisfies x < 1 or x >= 1. So the domain is (-∞, ∞).

• Finding the Range: This is a bit trickier. For x < 1, g(x) = x^2. Since x is less than 1, x^2 will be greater than or equal to 0 and less than 1. So on the interval (-∞, 1), the range is [0, 1). For x >= 1, g(x) = 2x + 1. Since x is greater than or equal to 1, 2x + 1 will be greater than or equal to 3. So on the interval [1, ∞), the range is [3, ∞). The overall range is therefore [0, 1) ∪ [3, ∞).

Example 3: Function Defined by a Table

Suppose the function g is defined by the following table:

• Function Evaluation: From the table, we can directly read off the values. For example, g(-2) = 4, g(0) = 0, and g(3) = 9.

• Finding the Domain: The domain consists of the x-values in the table: {-2, -1, 0, 1, 2, 3}.

• Finding the Range: The range consists of the g(x)-values in the table: {0, 1, 4, 9}.

Example 4: Function Defined Verbally

Suppose the function g is defined as follows: "g(x) is the absolute value of x minus 2."

This can be translated into an equation:

g(x) = |x - 2|

• Function Evaluation:

  g(0) = |0 - 2| = |-2| = 2
  g(2) = |2 - 2| = |0| = 0
  g(-3) = |-3 - 2| = |-5| = 5
  

• Finding the Domain: The absolute value function is defined for all real numbers, so the domain of g is all real numbers (-∞, ∞).

• Finding the Range: The absolute value function always returns a non-negative value. Therefore, the range of g is all non-negative real numbers [0, ∞).

Example 5: Function Composition

Let's say we have two functions:

f(x) = x + 2
g(x) = x^2

We want to find the composite function f(g(x)). This means we substitute g(x) into f(x):

f(g(x)) = f(x^2) = (x^2) + 2

So, f(g(x)) = x^2 + 2.

Now, let's find g(f(x)). This means we substitute f(x) into g(x):

g(f(x)) = g(x + 2) = (x + 2)^2 = x^2 + 4x + 4

So, g(f(x)) = x^2 + 4x + 4. Notice that f(g(x)) and g(f(x)) are generally different functions.

Example 6: Finding the Inverse Function

Suppose the function g is defined as follows:

g(x) = 2x + 3

To find the inverse function, g⁻¹(x), we follow these steps:

1. Replace g(x) with y:

   y = 2x + 3
   

2. Swap x and y:

   x = 2y + 3
   

3. Solve for y:

   x - 3 = 2y
   y = (x - 3) / 2
   

4. Replace y with g⁻¹(x):

   g⁻¹(x) = (x - 3) / 2
   

Therefore, the inverse function is g⁻¹(x) = (x - 3) / 2. To verify this, we can check that g(g⁻¹(x)) = x and g⁻¹(g(x)) = x:

g(g⁻¹(x)) = g((x-3)/2) = 2((x-3)/2) + 3 = (x - 3) + 3 = x
g⁻¹(g(x)) = g⁻¹(2x + 3) = ((2x + 3) - 3) / 2 = (2x) / 2 = x

Common Types of Functions

Understanding the characteristics of different types of functions is essential. Here are a few common ones:

• Linear Function: A function of the form g(x) = mx + b, where m and b are constants. The graph is a straight line.
• Quadratic Function: A function of the form g(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The graph is a parabola.
• Polynomial Function: A function of the form g(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n, a_{n-1}, ..., a_1, a_0 are constants and n is a non-negative integer.
• Rational Function: A function that is the ratio of two polynomials, g(x) = p(x) / q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0.
• Exponential Function: A function of the form g(x) = a^x, where a is a positive constant and a ≠ 1.
• Logarithmic Function: The inverse of an exponential function. A function of the form g(x) = log_a(x), where a is a positive constant and a ≠ 1.
• Trigonometric Functions: Functions such as sine (sin x), cosine (cos x), and tangent (tan x), which relate angles of a right triangle to ratios of its sides.
• Absolute Value Function: A function defined as g(x) = |x|, which returns the non-negative value of x.

Restrictions on the Domain

It's important to be aware of potential restrictions on the domain of a function. These restrictions arise when certain operations are undefined for particular input values. Common scenarios include:

• Division by Zero: A rational function p(x)/q(x) is undefined when q(x) = 0. Therefore, we must exclude any x-values that make the denominator zero from the domain.
• Square Roots of Negative Numbers: The square root function √x is only defined for non-negative real numbers. Therefore, we must ensure that the expression under the square root is greater than or equal to zero. This applies to any even root (4th root, 6th root, etc.).
• Logarithms of Non-Positive Numbers: The logarithmic function log_a(x) is only defined for positive real numbers. Therefore, we must ensure that the argument of the logarithm is strictly greater than zero.

Example illustrating Domain Restrictions:

Suppose the function g is defined as follows:

g(x) = √(x - 4) / (x - 7)

We have two potential restrictions:

1. Square Root: x - 4 must be greater than or equal to zero:

   x - 4 >= 0
   x >= 4
   

2. Division by Zero: x - 7 cannot be equal to zero:

   x - 7 ≠ 0
   x ≠ 7
   

Combining these restrictions, the domain of g is all real numbers greater than or equal to 4, except for 7. In interval notation, this is [4, 7) ∪ (7, ∞).

Transformations of Functions

Understanding how to transform functions can provide valuable insights into their behavior. Common transformations include:

• Vertical Shifts: Adding a constant to a function shifts its graph vertically. For example, g(x) + c shifts the graph of g(x) upward by c units if c > 0, and downward by |c| units if c < 0.
• Horizontal Shifts: Replacing x with x - c shifts the graph horizontally. For example, g(x - c) shifts the graph of g(x) to the right by c units if c > 0, and to the left by |c| units if c < 0.
• Vertical Stretching/Compression: Multiplying a function by a constant stretches or compresses its graph vertically. For example, a * g(x) stretches the graph vertically by a factor of a if a > 1, and compresses it vertically by a factor of a if 0 < a < 1. If a is negative, it also reflects the graph across the x-axis.
• Horizontal Stretching/Compression: Replacing x with ax stretches or compresses the graph horizontally. For example, g(ax) compresses the graph horizontally by a factor of a if a > 1, and stretches it horizontally by a factor of a if 0 < a < 1. If a is negative, it also reflects the graph across the y-axis.
• Reflection across the x-axis: Multiplying the entire function by -1 reflects the graph across the x-axis: -g(x).
• Reflection across the y-axis: Replacing x with -x reflects the graph across the y-axis: g(-x).

Applications of Functions

Functions are fundamental to many areas of mathematics, science, and engineering. They are used to model a wide variety of phenomena, including:

• Physics: Describing the motion of objects, the relationship between force and acceleration, and the behavior of electromagnetic waves.
• Economics: Modeling supply and demand curves, cost functions, and revenue functions.
• Computer Science: Defining algorithms, data structures, and programming languages.
• Statistics: Representing probability distributions and statistical models.
• Engineering: Designing circuits, structures, and control systems.

Conclusion

Understanding the definition, representation, and properties of functions is essential for success in mathematics and related fields. Whether defined by an equation, graph, table, or verbal description, a function provides a powerful tool for modeling and analyzing relationships between variables. By mastering the concepts discussed, from function evaluation to domain restrictions and transformations, you will be well-equipped to tackle a wide range of mathematical problems and real-world applications involving functions. The function g, regardless of its specific definition, represents a foundational concept upon which more advanced mathematical ideas are built. Keep practicing and exploring different types of functions to deepen your understanding!