Compare y=1/(x+3)+3 and y=1/x
To compare the two functions, we can start by analyzing their characteristics, such as their domain, range, intercepts, asymptotes, and behaviors.
Let's start with the first function, y = 1/(x + 3) + 3:
1. Domain: The function is defined for all real values of x except when the denominator (x + 3) is equal to zero. Thus, the function is defined for all x ≠ -3.
2. Range: As x approaches positive infinity or negative infinity, the function approaches 3. Therefore, the range of the function is all real numbers except 3.
3. Intercepts: To find the x-intercept, we set y = 0 and solve for x. In this case, it is not possible to find an x-intercept since the function is always positive. To find the y-intercept, we set x = 0 and calculate y. By substituting x = 0 into the equation, we get y = 1/3. Therefore, the y-intercept is (0, 1/3).
4. Asymptotes: This function has two asymptotes, a vertical asymptote and a horizontal asymptote. The vertical asymptote occurs when the denominator is equal to zero. So, x + 3 = 0 → x = -3. Therefore, the vertical asymptote is x = -3. The horizontal asymptote can be found by comparing the degrees of the leading terms in the numerator and denominator. Since both have the same degree (1), the horizontal asymptote is y = 0.
5. Behavior: As x approaches positive infinity or negative infinity, the function approaches the horizontal asymptote y = 0. As x approaches -3 from the left or the right, the function approaches positive infinity.
Now, let's analyze the second function, y = 1/x:
1. Domain: The function is defined for all real values of x except when x equals zero. Therefore, the domain of this function is all real numbers except 0.
2. Range: As x approaches positive infinity or negative infinity, the function approaches 0. Therefore, the range of the function is all real numbers except 0.
3. Intercepts: To find the x-intercept, we set y = 0 and solve for x. In this case, it is not possible to find an x-intercept since the function is never equal to zero. To find the y-intercept, we set x = 0 and calculate y. By substituting x = 0 into the equation, we get y = undefined. Therefore, there is no y-intercept.
4. Asymptotes: This function has two asymptotes, a vertical asymptote and a horizontal asymptote. The vertical asymptote occurs when the denominator is equal to zero. So, x = 0. Therefore, the vertical asymptote is the line x = 0. The horizontal asymptote can be found by comparing the degrees of the leading terms in the numerator and denominator. Since the degree of the numerator is 0 (constant), and the degree of the denominator is 1, the horizontal asymptote is y = 0.
5. Behavior: As x approaches positive infinity or negative infinity, the function approaches the horizontal asymptote y = 0. As x approaches 0 from the left or the right, the function approaches positive infinity or negative infinity, respectively.
In summary, these two functions have different properties. The first function, y = 1/(x + 3) + 3, has a vertical asymptote at x = -3 and a horizontal asymptote at y = 0. The second function, y = 1/x, has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The two functions also have different domains, ranges, and intercepts.