Find the domain, range and vertical asymptotes for the following problems:
f(x)=2^x-3, f(x)=1+3^x, f(x)=ln(x-1)-2
To find the domain, range, and vertical asymptotes of the given functions, we need to analyze each function separately. Let's take a look at each function one by one:
1. f(x) = 2^x - 3:
- Domain: The domain for this function is all real numbers since there are no restrictions on the base (2) and the exponent (x).
- Range: The range of the function depends on the behavior of the exponential function. Since the base of the exponential function, 2, is positive, the function will always be positive. Therefore, the range is all real numbers greater than or equal to -3.
- Vertical Asymptotes: Exponential functions do not have vertical asymptotes, so there are no vertical asymptotes for this function.
2. f(x) = 1 + 3^x:
- Domain: Similar to the previous function, the domain for this function is all real numbers.
- Range: The range of the function depends on the behavior of the exponential function. Since the base of the exponential function, 3, is positive, the function will always be positive. Therefore, the range is all real numbers greater than or equal to 1.
- Vertical Asymptotes: Exponential functions do not have vertical asymptotes, so there are no vertical asymptotes for this function as well.
3. f(x) = ln(x - 1) - 2:
- Domain: The domain of the function is determined by the values that x can take which keep the logarithm defined. In this case, x must be greater than 1 because the natural logarithm (ln) is undefined for non-positive values. Therefore, the domain is all real numbers greater than 1.
- Range: The range of the natural logarithm function, ln(x), is all real numbers. By subtracting 2 from the function, the range will shift downward by 2 units. Therefore, the range of the given function is all real numbers less than or equal to -2.
- Vertical Asymptotes: The natural logarithm function has a vertical asymptote at x = 0. However, in this case, the function is shifted 1 unit to the right, so the vertical asymptote occurs at x = 1.
In summary:
1. f(x) = 2^x - 3:
Domain: all real numbers
Range: all real numbers greater than or equal to -3
Vertical Asymptotes: none
2. f(x) = 1 + 3^x:
Domain: all real numbers
Range: all real numbers greater than or equal to 1
Vertical Asymptotes: none
3. f(x) = ln(x - 1) - 2:
Domain: all real numbers greater than 1
Range: all real numbers less than or equal to -2
Vertical Asymptotes: x = 1