Question
The domain of a reciprocal function is (−∞,4)∪(4,∞) . It's horizontal asymptote is at y=5 . What is the equation of the function? Explain how you know.(4 points)
Answers
GPT-4o mini
To find the equation of a reciprocal function with the provided characteristics, we can start with the general form of a reciprocal function, which is typically given as:
\[
f(x) = \frac{k}{x - h} + b
\]
Where:
- \(k\) is a constant that affects the vertical stretch of the function,
- \(h\) is the horizontal shift,
- \(b\) is the vertical shift.
Given that the domain of the function is \((-\infty, 4) \cup (4, \infty)\), we see that there is a vertical asymptote at \(x = 4\). This indicates that the function approaches infinity as \(x\) approaches 4 from the left and negative infinity as \(x\) approaches 4 from the right. This leads us to set \(h = 4\).
Next, we know that the horizontal asymptote is at \(y = 5\). This means that as \(x\) approaches either positive or negative infinity, the function \(f(x)\) will approach 5. In the general form of the function, we can interpret this by rewriting the expression as:
\[
f(x) = \frac{k}{x - 4} + 5
\]
As \(x\) goes to \(±\infty\), the term \(\frac{k}{x - 4}\) approaches 0, leaving us with:
\[
\lim_{x \to \pm\infty} f(x) = 5
\]
This confirms that the horizontal asymptote of the function is \(y = 5\).
Now, \(k\) can be any non-zero value, as it determines the steepness or rate of change of the function, but does not affect the asymptotes. For simplicity, let’s choose \(k = 1\). So, the equation of the function can be expressed as:
\[
f(x) = \frac{1}{x - 4} + 5
\]
### Summary
1. The vertical asymptote is at \(x = 4\) (vertical line).
2. The horizontal asymptote is at \(y = 5\).
3. The chosen function, \(f(x) = \frac{1}{x - 4} + 5\), satisfies both asymptotic conditions.
Thus, the equation of the function is:
\[
f(x) = \frac{1}{x - 4} + 5.
\]
\[
f(x) = \frac{k}{x - h} + b
\]
Where:
- \(k\) is a constant that affects the vertical stretch of the function,
- \(h\) is the horizontal shift,
- \(b\) is the vertical shift.
Given that the domain of the function is \((-\infty, 4) \cup (4, \infty)\), we see that there is a vertical asymptote at \(x = 4\). This indicates that the function approaches infinity as \(x\) approaches 4 from the left and negative infinity as \(x\) approaches 4 from the right. This leads us to set \(h = 4\).
Next, we know that the horizontal asymptote is at \(y = 5\). This means that as \(x\) approaches either positive or negative infinity, the function \(f(x)\) will approach 5. In the general form of the function, we can interpret this by rewriting the expression as:
\[
f(x) = \frac{k}{x - 4} + 5
\]
As \(x\) goes to \(±\infty\), the term \(\frac{k}{x - 4}\) approaches 0, leaving us with:
\[
\lim_{x \to \pm\infty} f(x) = 5
\]
This confirms that the horizontal asymptote of the function is \(y = 5\).
Now, \(k\) can be any non-zero value, as it determines the steepness or rate of change of the function, but does not affect the asymptotes. For simplicity, let’s choose \(k = 1\). So, the equation of the function can be expressed as:
\[
f(x) = \frac{1}{x - 4} + 5
\]
### Summary
1. The vertical asymptote is at \(x = 4\) (vertical line).
2. The horizontal asymptote is at \(y = 5\).
3. The chosen function, \(f(x) = \frac{1}{x - 4} + 5\), satisfies both asymptotic conditions.
Thus, the equation of the function is:
\[
f(x) = \frac{1}{x - 4} + 5.
\]