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Magazine Chapter 7: Problem 45
Find the range of the given function, and express your answer in set notation. \(f(x)=\frac{4}{x+3}-2\)
Short Answer
Expert verified
The range of the function is \(( -\infty, -2 ) \cup ( -2, \infty )\).
Step by step solution
01
Identify the Domain Restrictions
For the function \( f(x) = \frac{4}{x+3} - 2 \), note that the denominator \( x+3 \) cannot be zero because division by zero is undefined. Therefore, \( x eq -3 \).
02
Understand the Function Behavior
The function \( f(x) = \frac{4}{x+3} - 2 \) is derived from the parent function \( y = \frac{1}{x} \). It is transformed by a horizontal shift 3 units to the left, a vertical stretch by a factor of 4, followed by a vertical shift downward by 2 units.
03
Identify Horizontal Asymptote
As \( x \to \infty \) or \( x \to -\infty \), the \( \frac{4}{x+3} \) term approaches \( 0 \). Thus, the function approaches \( f(x) = -2 \). Hence, the horizontal asymptote is \( y = -2 \), which indicates the function never equals \(-2\) but can get infinitely close.
04
Determine the Range
Since the \( \frac{4}{x+3} \) term can take any real positive or negative value except that which makes the denominator zero, \( f(x) \) itself can take any real value except \(-2\). Therefore, the range of \( f(x) \) is all real numbers except \(-2\), expressed in set notation as: \(( -\infty, -2 ) \cup ( -2, \infty )\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Domain Restrictions
Understanding domain restrictions is a crucial first step in analyzing functions. The domain of a function includes all the possible input values (x-values) for which the function is defined. In the case of the function \( f(x) = \frac{4}{x+3} - 2 \), it is necessary to consider the expression in the denominator \( x+3 \).
Division by zero is undefined in mathematics; thus, we must identify values of \( x \) that lead to a zero in the denominator, as these are not allowed.
Division by zero is undefined in mathematics; thus, we must identify values of \( x \) that lead to a zero in the denominator, as these are not allowed.
- Set \( x+3 = 0 \) to find the restriction, solving leads to \( x = -3 \).
- Therefore, the domain excludes \( x = -3 \) and includes all other real numbers.
Horizontal Asymptote
Horizontal asymptotes provide insight into the behavior of a function as the input values grow very large in magnitude (either positively or negatively). For the function \( f(x) = \frac{4}{x+3} - 2 \), we look at how \( f(x) \) behaves as \( x \to \infty \) or \( x \to -\infty \).
- The term \( \frac{4}{x+3} \) becomes very small as \( x \) grows larger, effectively approaching zero.
- This implies that \( f(x) \) approaches \( -2 \) but never actually reaches it.
Function Transformations
Function transformations help us understand how to manipulate a basic function into a more complex form. The given function \( f(x) = \frac{4}{x+3} - 2 \) derives from the basic function \( y = \frac{1}{x} \).
Here鈥檚 a breakdown of the transformations:
Here鈥檚 a breakdown of the transformations:
- **Horizontal Shift**: Moving 3 units to the left, from \( y = \frac{1}{x} \) to \( y = \frac{1}{x+3} \).
- **Vertical Stretch**: Factor of 4 stretching the function vertically, leading to \( y = \frac{4}{x+3} \).
- **Vertical Shift**: Finally, shifting the entire graph 2 units down results in \( f(x) = \frac{4}{x+3} - 2 \).
Set Notation
Set notation is a powerful way to express the domain or range of a function in a precise manner. For the function \( f(x) = \frac{4}{x+3} - 2 \), we've identified its range, meaning all possible output values except any for which the function isn't defined.
The function never truly reaches \( -2 \) due to the horizontal asymptote.
Therefore, while it can get infinitely close to \( -2 \), it stays clear of it.
The function never truly reaches \( -2 \) due to the horizontal asymptote.
Therefore, while it can get infinitely close to \( -2 \), it stays clear of it.
- This translates into a range that includes every real number except \( -2 \).
- Using set notation, this is expressed as \(( -\infty, -2 ) \cup ( -2, \infty )\).
