Question

Consider \(f(x)=x^{2}-9\) and \(g(x)=\frac{1}{x}\) a) State the domain and range of each function. b) Determine \(h(x)=f(x)+g(x)\). c) How do the domain and range of each function compare to the domain and range of \(h(x) ?\).

Short answer

The domain of \( h(x) \) is \( x eq 0 \). The range of \( h(x) \) is all real numbers.

Step-by-step solution

Identify the Domain and Range of f(x)

The function given is \( f(x) = x^2 - 9 \). This is a parabola that opens upwards. Domain: All real numbers (\( -\infty, \infty \)). Range: All values \( y \geq -9 \).

Identify the Domain and Range of g(x)

The function given is \( g(x) = \frac{1}{x} \). This is a rational function. Domain: All real numbers except zero (\( x eq 0 \)). Range: All real numbers except zero (\( y eq 0 \)).

Determine h(x) = f(x) + g(x)

The new function is \( h(x) = f(x) + g(x) = x^2 - 9 + \frac{1}{x} \). Combine both functions into one expression.

Find the Domain of h(x)

Since \( h(x) \) is the sum of \( f(x) \) and \( g(x) \), the domain of \( h(x) \) is the intersection of the domains of both functions. Therefore, the domain is all real numbers except zero (\( x eq 0 \)).

Find the Range of h(x)

To find the range of \( h(x) \), observe that as \( x \) approaches zero from both positive and negative sides, \( \frac{1}{x} \) becomes very large in magnitude, either positive or negative. Away from zero, the quadratic term \( x^2 - 9 \) dominates. This analysis hints that the range of \( h(x) \) is all real numbers (\( y \in \mathbb{R} \)).

Compare the Domain and Range of Each Function

The domain of \( f(x) \) is all real numbers, while \( g(x) \) and \( h(x) \) have domain \( x eq 0 \). The range of \( f(x) \) is \( y \geq -9 \), the range of \( g(x) \) is \( y eq 0 \), and the range of \( h(x) \) is all real numbers.

Key concepts

Domain and Range

Before diving into the specifics of the problem, it's essential to grasp the concepts of domain and range. The domain of a function specifies all the possible input values (x-values) that the function can accept. Conversely, the range of a function describes all the possible output values (y-values) produced by the function.
For example, for the function \( f(x) = x^2 - 9 \), the domain is all real numbers because a parabola extends infinitely in both left and right directions. So, the domain is \( -oty otx otx otx otx otx otx otx otx otx otx otx otx oty \). For the range, the smallest value is -9, since the quadratic function reaches its minimum at the vertex, yielding the range \( y otx \frac{1}{x} \) is all real numbers except zero. Any number divided by zero is undefined. Consequently, the domain is considered as \( -oty < x < 0 otx 0 < x < oty otx oty oty \) covers all real numbers except zero.

Quadratic Functions

A quadratic function is any function that can be expressed in the form \( f(x) = ax^2 + bx + c \), where a, b, and c are constants and \( a eq 0 \). They produce parabolas when graphed.
In our given exercise, \(f(x) = x^2 - 9\). The graph of this function is a parabola that opens upwards because the coefficient of \( x^2 \) is positive.
Key features of quadratic functions include:

• Vertex: The highest or lowest point on the graph of the function. For \( f(x) = x^2 - 9 \), the vertex is at (0, -9).
• Axis of Symmetry: A vertical line that splits the parabola into two symmetrical halves. For the function \( f(x) = x^2 - 9 \), the axis of symmetry is the y-axis or \( x = 0 \).
• Roots or Zeros: The x-values where the function equals zero. For \( f(x) = x^2 - 9 \), the roots are 3 and -3, solving the equation \( x^2 - 9 = 0 \).

Understanding these components can help you graph and analyze quadratic functions more effectively.

Rational Functions

Rational functions are ratios of two polynomials, represented as \( g(x) = \frac{P(x)}{Q(x)} \), where both P(x) and Q(x) are polynomials and \( Q(x) eq 0 \). These functions can have asymptotes and holes because the denominator cannot equal zero.
In the exercise, \( g(x) = \frac{1}{x} \). It's a basic rational function with a vertical asymptote at \( x = 0 \) and a horizontal asymptote along the x-axis \( y = 0 \).
Important aspects to remember are:

• Vertical Asymptote: Happens when the denominator equals zero. For example, in \( g(x) = \frac{1}{x} \), the vertical asymptote occurs at \( x = 0 \).
• Horizontal Asymptote: Describes the behavior as x approaches positive or negative infinity. Here, \( g(x) \) trends towards zero as x increases or decreases infinitely.

Rational functions can be more challenging to graph due to these asymptotes and non-continuous behavior. Keep these points in mind to better understand and manipulate rational functions.

Combining Functions

Combining functions involves creating a new function by performing operations like addition, subtraction, multiplication, or division on two functions. In our exercise, we're adding functions \( f(x) = x^2 - 9 \) and \( g(x) = \frac{1}{x} \) to form \( h(x) = f(x) + g(x) = x^2 - 9 + \frac{1}{x} \).
When combining functions, it's crucial to:

• Consider the Domain: The domain of the resulting function \( h(x) \) is determined by the intersection of the domains of \( f(x) \) and \( g(x). \) Here, both functions are defined for all real numbers except zero, hence the domain of \( h(x) \) is also all real numbers except zero.
• Determine the Range: Analyzing the range of the new function can be complex. For \( h(x) \), as x approaches zero, \( \frac{|1|}{x} \) becomes very large, covering all y-values; thus, the range of \( h(x) \) is all real numbers, \( \mathbb{R} \).

Combining functions is a valuable tool in understanding more complex behaviors by leveraging simpler individual functions. Always pay attention to the resulting domain and range to ensure the new function's validity.