Graph y=5^x and y=log_5 x on a sheet of paper using the same set of axes. Use the graph to describe the domain and range of each function. Then identify the y-intercept of each function and any asymptotes of each function.
So far I have:
Y=5^x :
D: ??
R: y>0
Y-intercept: 1
Asymptotes: y=0
__________________________________
Y=log_5 x :
D:??
R:??
Y-intercept: none
Asymptotes: x=0
can anyone tell me if this is correct? Or help me find out how to do the other parts that I was not able to find out? Thank you so much!!
the domain of all exponentials is (-∞,∞)
and the range is (0,∞)
For logs, they are reversed, since it is the inverse function.
exponentials all pass through (0,1)
logs all pass through (1,0)
your asymptotes are also correct.
So for the first one:
Domain: (-∞,∞)
Range: (0,∞)
Y-intercept: 1
and for the second one:
Domain: (∞,-∞)
Range: (∞,0)
Y-intercept: -1
this is what I understood from your response?
yeah! for my sake!
why not my sake? >:(
Sure! Let's break it down:
1. For the function y = 5^x:
- Domain (D): Since the base is positive, the domain is all real numbers.
- Range (R): The range is all positive real numbers (y > 0) because 5^x is always positive.
- Y-intercept: The y-intercept occurs when x = 0. If we substitute x = 0 into the equation, we get y = 5^0 = 1. So, the y-intercept is (0, 1).
- Asymptote: There is no asymptote for this function.
2. For the function y = log_5(x):
- Domain (D): For the logarithmic function, the argument (x) must be positive, so the domain is x > 0.
- Range (R): The range of the logarithmic function is all real numbers.
- Y-intercept: The y-intercept occurs when x = 1. Substituting x = 1 into the equation, we get y = log_5(1) = 0. So, the y-intercept is (1, 0).
- Asymptote: The vertical asymptote occurs when x approaches 0 from the positive side because log_5(x) approaches negative infinity as x approaches 0. Therefore, the asymptote is x = 0.
Your understanding of the y-intercepts and asymptotes is correct, but here are the missing details for the domain and range of each function. I hope this helps! Let me know if you have any other questions.
Your initial information for the graph of y = 5^x is correct. Now let's move on to finding the domain and range of the function y = log_5(x).
To determine the domain, we need to consider the values that x can take. In this case, the logarithm function is defined only for positive values of x. Therefore, the domain of y = log_5(x) is x > 0.
For the range, we need to consider the values that y can take. The logarithm function returns the exponent to which the base (in this case 5) must be raised to give x. Since 5 raised to any positive exponent gives a positive value, the range of y = log_5(x) is y ∈ R (all real numbers).
Now let's move on to the y-intercept and asymptotes for the function y = log_5(x).
For the y-intercept, we need to find the value of y when x = 0. However, since the function y = log_5(x) is undefined for x = 0, there is no y-intercept for this function.
As for the asymptote, the logarithmic function has a vertical asymptote at x = 0. This means that as x approaches 0 from the positive side, the function value approaches negative infinity.
Therefore, the updated information for y = log_5(x) is as follows:
- Domain: x > 0
- Range: y ∈ R (all real numbers)
- Y-intercept: none
- Asymptotes: x = 0 (vertical asymptote)
Make sure to plot both functions on the same set of axes correctly, and you will have completed the task.
No, for logs, you have
Domain: (0,∞)
Range: (-∞,∞)
x-intercept: 1
Take a look at the graphs, fer pete's sake!