Domain, Vertical & Horizontal Asymptotes Of F(x) = 7/(x+6)

by Dimemap Team 59 views

Hey guys! Let's break down how to find the domain, vertical asymptotes, and horizontal asymptotes of the function f(x) = 7/(x+6). Understanding these concepts is super important in calculus and pre-calculus, so let's get started!

Domain of f(x) = 7/(x+6)

The domain of a function is simply the set of all possible input values (x-values) for which the function is defined. In other words, it's all the x-values that you can plug into the function without causing any mathematical mayhem, like dividing by zero or taking the square root of a negative number (in the realm of real numbers, of course!).

For our function, f(x) = 7/(x+6), we need to watch out for division by zero. Division by zero is a big no-no in mathematics because it leads to undefined results. So, we need to figure out what value(s) of x would make the denominator, x + 6, equal to zero.

To find these values, we set the denominator equal to zero and solve for x:

x + 6 = 0

Subtracting 6 from both sides, we get:

x = -6

This tells us that when x = -6, the denominator x + 6 becomes zero, and our function becomes undefined. Therefore, x = -6 is not in the domain of the function.

So, the domain of f(x) = 7/(x+6) is all real numbers except x = -6. We can express this in a few different ways:

  • Set Notation: {x | x ∈ ℝ, x ≠ -6}
  • Interval Notation: (-∞, -6) ∪ (-6, ∞)

In interval notation, we use parentheses to indicate that -6 is not included in the domain. The union symbol (∪) combines the two intervals, representing all real numbers less than -6 and all real numbers greater than -6.

Therefore, the domain is all real numbers except -6, because plugging in -6 would make the denominator zero, and that's a big no-no in math town! Remember, always be on the lookout for potential division by zero scenarios when determining the domain of a rational function. Identifying the domain is the crucial first step before diving into asymptotes.

Vertical Asymptote(s) of f(x) = 7/(x+6)

A vertical asymptote is a vertical line that the graph of a function approaches but never actually touches or crosses. Vertical asymptotes occur at x-values where the function becomes unbounded (i.e., approaches infinity or negative infinity). Think of it as the function getting super close to a certain x-value, but never quite making it.

In rational functions (functions that are fractions with polynomials), vertical asymptotes often occur where the denominator is equal to zero, provided that the numerator is not also zero at the same point. We already found that the denominator of f(x) = 7/(x+6) is zero when x = -6. Now we need to make sure the numerator isn't also zero at x = -6. Since the numerator is the constant 7, it's never zero, so we're good to go!

Since the denominator x + 6 is zero at x = -6, and the numerator is not zero at this point, there is a vertical asymptote at x = -6. The graph of the function will get closer and closer to the vertical line x = -6 as x approaches -6 from the left and from the right, but it will never actually touch or cross it.

To confirm this, you could analyze the limits as x approaches -6 from both sides:

  • lim (x→-6-) f(x) = -∞ (As x approaches -6 from the left, the function goes to negative infinity)
  • lim (x→-6+) f(x) = +∞ (As x approaches -6 from the right, the function goes to positive infinity)

The presence of these infinite limits confirms that there is a vertical asymptote at x = -6. Remember, vertical asymptotes are all about those spots where the function goes wild and shoots off to infinity (or negative infinity). Finding where the denominator equals zero (and the numerator doesn't) is your golden ticket to finding those vertical asymptotes. So, in this case, x = -6 is our vertical asymptote. Vertical asymptotes are key to understanding the behavior of rational functions around specific x-values.

Horizontal Asymptote of f(x) = 7/(x+6)

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity. In simpler terms, it's what value y approaches as x gets super, super big (positive or negative). Think of it as a long-term trend – what value does the function settle down to as x goes way out to the edges of the graph?

To find the horizontal asymptote of a rational function, we compare the degrees of the polynomials in the numerator and denominator. The degree of a polynomial is the highest power of x in the polynomial.

In our function, f(x) = 7/(x+6):

  • The degree of the numerator (7) is 0 (since 7 can be thought of as 7x⁰).
  • The degree of the denominator (x+6) is 1.

Since the degree of the numerator (0) is less than the degree of the denominator (1), the horizontal asymptote is y = 0. This means that as x gets very large (positive or negative), the value of the function f(x) gets closer and closer to 0.

In general, here's the rule for finding horizontal asymptotes of rational functions:

  • If the degree of the numerator is less than the degree of the denominator: The horizontal asymptote is y = 0.
  • If the degree of the numerator is equal to the degree of the denominator: The horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator).
  • If the degree of the numerator is greater than the degree of the denominator: There is no horizontal asymptote (but there might be a slant asymptote).

In our case, since 0 < 1, we have a horizontal asymptote at y = 0. This means that as x heads off to infinity or negative infinity, the function's value gets closer and closer to zero. Horizontal asymptotes describe the long-term behavior of functions and give a sense of where the graph is heading as x moves toward the extremes. It's like the function is trying to cozy up to the x-axis as it goes far away from the origin.

Summary

Alright, to recap, for the function f(x) = 7/(x+6), we found:

  • Domain: All real numbers except x = -6 (or (-∞, -6) ∪ (-6, ∞) in interval notation).
  • Vertical Asymptote: x = -6
  • Horizontal Asymptote: y = 0

Understanding these key features helps you sketch the graph of the function and understand its behavior. You got this!