Hi all, hope someone could help me. I'm trying to find domain, range and asymptotes for the following functions.
f(x) = 5 - e^(x-1)
g(x) = ln(5 - y) + 1
The variable on the right side of g(x) should be x, not y.
Take a close look at the functions. Plotting some points may help.
Review the definitions of domain, range and asymptote and tell us what you think.
As for the domain: Are any values of x not allowed?
Hint: You cannot take the log of a negative number or zero.
Sorry, typo...
To determine the domain, range, and asymptotes for the given functions, let's analyze each function separately.
For the function f(x) = 5 - e^(x-1):
1. Domain: The domain of a function refers to all the possible values of x for which the function is defined. In this case, the exponential function e^(x-1) is defined for all real numbers. Therefore, there are no restrictions on the domain of f(x), and it extends to all real numbers (-∞, +∞).
2. Range: The range of a function refers to all the possible values of y that the function can produce. In this case, as the exponential function e^(x-1) is always positive, it means that f(x) = 5 - e^(x-1) can never be greater than 5 nor less than negative infinity. Therefore, the range of f(x) is (-∞, 5].
3. Asymptotes: Asymptotes are a line that the graph of a function approaches but never intersects. Since we're dealing with an exponential function, there are no horizontal asymptotes. However, there is a vertical asymptote at x = 1 because when x approaches 1 from the left side, e^(x-1) approaches infinity. In other words, the function f(x) approaches positive infinity as x approaches 1 from the left side.
Now, let's move on to the function g(x) = ln(5 - y) + 1:
1. Domain: The domain of g(x) is determined by the argument of the natural logarithm. For the function ln(5 - y), the argument (5 - y) must be greater than zero since the natural logarithm is only defined for positive numbers. Therefore, we need the inequality 5 - y > 0 to determine the domain. Solving this inequality, we find that y < 5. Thus, the domain of g(x) is (-∞, 5).
2. Range: The range of g(x) represents the values that the function can produce. Since the natural logarithm is defined for all positive numbers, there are no restrictions on the range of g(x). Therefore, the range of g(x) is (-∞, +∞).
3. Asymptotes: For the function g(x) = ln(5 - y) + 1, there are no vertical asymptotes as the logarithmic function does not have vertical asymptotes. However, there is a horizontal asymptote at y = 1 because as y approaches positive or negative infinity, ln(5 - y) approaches negative infinity, and when we add 1, it becomes y = 1. Thus, the function g(x) approaches y = 1 as y approaches positive or negative infinity.
In summary, for the function f(x) = 5 - e^(x-1):
- Domain: (-∞, +∞)
- Range: (-∞, 5]
- Vertical asymptote: x = 1
For the function g(x) = ln(5 - y) + 1:
- Domain: (-∞, 5)
- Range: (-∞, +∞)
- Horizontal asymptote: y = 1