Determine The Domain And Range Of (g F)(x) If F Of X Is Equal To 9 Over The Quantity X Squared Minus
Understanding the domain and range of functions is fundamental in mathematics, especially when analyzing composite functions such as (g ◦ F)(x). The process involves examining how the inner function F(x) behaves and how it influences the overall behavior of the composite function. In this article, we will explore the steps to determine the domain and range of (g ◦ F)(x) when F(x) is given as 9 over the quantity x squared minus a constant, which is a common form in rational functions.
This topic is crucial for students and professionals working in fields related to mathematics, engineering, physics, and economics, where understanding the possible input and output values of functions helps in modeling real-world scenarios accurately. By delving into the specific case where F(x) = 9 / (x^2 - c), we will learn how to handle the restrictions imposed by denominators and how to analyze the resulting composite function comprehensively.
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Understanding the Components of the Problem
What Is a Composite Function?
A composite function, denoted as (g ◦ F)(x), is formed by applying one function, g, to the output of another function, F. Mathematically, it is expressed as:
\[ (g ◦ F)(x) = g(F(x)) \]
To analyze the domain and range of the composite, we need to understand both the inner function F(x) and the outer function g(x). The overall domain depends on the set of x-values for which F(x) is defined and for which g(F(x)) is also defined. Similarly, the range depends on the possible outputs of the composite function.
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Examining the Inner Function F(x) = 9 / (x^2 - c)
Properties of the Rational Function
The given inner function is a rational function:
\[ F(x) = \frac{9}{x^2 - c} \]
where c is a constant (the specific value is not provided but can be any real number). The key features to analyze are:
- Denominator restrictions: Since division by zero is undefined, the denominator cannot be zero:
\[ x^2 - c \neq 0 \Rightarrow x^2 \neq c \Rightarrow x \neq \pm \sqrt{c} \]
- Behavior at critical points: The function approaches infinity or negative infinity as x approaches ±√c, indicating vertical asymptotes at these points.
- Domain of F(x): All real numbers except x = ±√c.
Domain of F(x):
\[ \boxed{\text{All real } x \text{ such that } x \neq \pm \sqrt{c}} \]
Range of F(x):
Since \( F(x) = \frac{9}{x^2 - c} \), and as \( x^2 \) varies over [0, ∞), the denominator \( x^2 - c \) varies over \( (-c, \infty) \) (excluding zero). The numerator is positive (9), so:
- When \( x^2 - c \to 0^+ \), \( F(x) \to +\infty \).
- When \( x^2 - c \to 0^- \), \( F(x) \to -\infty \).
- As \( x^2 \to \infty \), \( F(x) \to 0^+ \).
Therefore, the range depends on the value of c.
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Determining the Range of F(x) Based on c
The behavior of F(x) is influenced by whether c is positive, negative, or zero.
Case 1: c > 0
- The vertical asymptotes are at \( x = \pm \sqrt{c} \).
- For \( x^2 > c \), denominator \( x^2 - c \) is positive, so \( F(x) > 0 \).
- For \( x^2 < c \), the denominator is negative, so \( F(x) < 0 \).
- As \( x \to \pm \sqrt{c}^- \), \( F(x) \to -\infty \).
- As \( x \to \pm \sqrt{c}^+ \), \( F(x) \to +\infty \).
- As \( x \to \pm \infty \), \( F(x) \to 0^+ \).
\[ (-\infty, 0) \cup (0, \infty) \]
excluding zero because F(x) approaches zero but never reaches it.
Case 2: c < 0
- Vertical asymptotes at \( x = \pm \sqrt{c} \), but since c is negative, \( \sqrt{c} \) is imaginary, so no real vertical asymptotes.
- The denominator \( x^2 - c \) is always positive because:
for all real x.
- As \( x \to \pm \infty \), \( F(x) \to 0^+ \).
- The minimum of \( x^2 - c \) is at \( x=0 \), where:
\[ F(0) = \frac{9}{-c} \]
Since c < 0, \( -c > 0 \), so \( F(0) > 0 \).
- The maximum of F(x) occurs at \( x=0 \):
\[ F(0) = \frac{9}{-c} \]
- As \( x \to \pm \infty \), \( F(x) \to 0^+ \).
Range of F(x) for c < 0:
\[ (0, \frac{9}{-c}) \]
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Determining the Domain of the Composite Function (g ◦ F)(x)
To find the domain of \( (g ◦ F)(x) \), we must consider:
- The domain of F(x): all real x except where the denominator is zero.
- The domain of g(y): all y in the range of F(x) for which g(y) is defined.
Suppose g(y) is defined for all real y, or at least for a subset of y-values.
Step-by-step process:
- Step 1: Identify the domain of F(x):
- For c > 0: all real x except \( \pm \sqrt{c} \).
- For c < 0: all real x.
- Step 2: Determine the range of F(x), as discussed in the previous section.
- Step 3: Find the set of y-values where g(y) is defined, which is the intersection of g's domain with the range of F(x).
- Step 4: The domain of \( (g ◦ F)(x) \) is the set of all x in the domain of F(x) such that F(x) is in g's domain.
This process often involves solving inequalities or restrictions based on g(y).
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Analyzing the Range of the Composite Function (g ◦ F)(x)
The range of the composite depends on both the range of F(x) and the behavior of g(y).
Key considerations:
- If g(y) is defined for all real y, then the range of \( (g ◦ F)(x) \) is simply g applied to the range of F(x).
- If g(y) has restrictions (e.g., g(y) is defined only for y > 0), then the range of \( (g ◦ F)(x) \) is the image of the intersection between the range of F(x) and the domain of g(y).
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Practical Examples and Applications
Example 1: g(y) = y^2
Suppose \( g(y) = y^2 \), which is defined for all real y.
- For c > 0:
- Range of F(x): \( (-\infty, 0) \cup (0, \infty) \).
- Applying g(y) = y^2:
- The image of F(x) under g is:
\[ \{ y^2 \mid y \neq 0 \} = [0, \infty) \]
- The domain of \( (g ◦ F)(x) \) is all x where F(x) ≠ 0, which is all real x except \( \pm \sqrt{c} \).
- The range of \( (g ◦ F)(x) \):
\[ [0, \infty) \]
- For c < 0:
- Range of F(x): \( (0, \frac{9}{-c}) \).
- Applying g(y) = y^2:
- The range of \( (g ◦ F)(x) \):
\[ (0, (\frac{9}{-c})^2) \]
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