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
Is this right?
Y=5^x
D: (-∞, ∞)
R: (0, ∞)
Y-intercept: 1
Asymptotes: y=0
Y=log_5 X
D: (0, ∞)
R: (-∞, ∞)
X-intercept: 1
Asymptotes: x=0
yes - note that they are exact inverses of each other.
To graph the functions y = 5^x and y = log_5(x) on the same set of axes, follow these steps:
Step 1: Draw the axes.
Start by drawing a vertical line (y-axis) and a horizontal line (x-axis) intersecting at the origin (0,0).
Step 2: Plotting points for y = 5^x.
For the function y = 5^x, we can start by plotting some points. For instance, let's consider x-values of -2, -1, 0, 1, and 2. Calculate the corresponding y-values by substituting the x-values into the equation.
For x = -2, y = 5^(-2) = 1/25 ≈ 0.04.
For x = -1, y = 5^(-1) = 1/5 ≈ 0.20.
For x = 0, y = 5^0 = 1.
For x = 1, y = 5^1 = 5.
For x = 2, y = 5^2 = 25.
Plot these points on the graph. (-2, 0.04), (-1, 0.20), (0, 1), (1, 5), and (2, 25).
Step 3: Plotting points for y = log_5(x).
For the function y = log_5(x), we can choose x-values of 1, 5, and 25 (since the base of the logarithm is 5). Calculate the corresponding y-values by taking the logarithm base 5 of each x-value.
For x = 1, y = log_5(1) = 0.
For x = 5, y = log_5(5) = 1.
For x = 25, y = log_5(25) = 2.
Plot these points on the graph. (1, 0), (5, 1), and (25, 2).
Step 4: Draw the graphs of the functions.
Connect the plotted points for each function. The graph of y = 5^x should be an increasing exponential curve that passes through all the plotted points, while the graph of y = log_5(x) should be a decreasing logarithmic curve passing through the points.
Step 5: Analyzing the domains and ranges.
For y = 5^x:
- Domain: The domain of 5^x includes all real numbers because any real number can be used as an exponent.
- Range: The range of 5^x is all positive real numbers because raising 5 to any power will always yield a positive value.
For y = log_5(x):
- Domain: The domain of log_5(x) includes all positive real numbers because the logarithm function only accepts positive values as input.
- Range: The range of log_5(x) is all real numbers because the logarithm function can output any real number.
Step 6: Identifying the y-intercept and asymptotes.
- For y = 5^x, the y-intercept is (0, 1) because 5^0 = 1. There are no vertical or horizontal asymptotes.
- For y = log_5(x), there is no y-intercept. The vertical asymptote is x = 0 because log_5(x) is undefined for x ≤ 0. There are no horizontal asymptotes.
With these steps, you should have the graph of both functions and the understanding of their domains, ranges, y-intercepts, and asymptotes.
To graph the functions y = 5^x and y = log_5 x, you can start by creating a set of axes on a sheet of paper. Label the x-axis and y-axis with appropriate scales.
Now, let's plot the graph of y = 5^x. To do this, choose some x-values and calculate the corresponding y-values using the exponential function. For example, let's choose x = -2, -1, 0, 1, and 2.
When x = -2, y = 5^(-2) = 1/25 ≈ 0.04.
When x = -1, y = 5^(-1) = 1/5 ≈ 0.2.
When x = 0, y = 5^0 = 1.
When x = 1, y = 5^1 = 5.
When x = 2, y = 5^2 = 25.
Plot these points on the graph, and you will notice that as x increases, the y-values increase rapidly. Draw a smooth curve passing through these points.
Now let's plot the graph of y = log_5 x. To do this, choose some positive x-values and calculate the corresponding y-values using the logarithmic function. For example, let's choose x = 1, 5, 25.
When x = 1, y = log_5 1 = 0.
When x = 5, y = log_5 5 = 1.
When x = 25, y = log_5 25 = 2.
Plot these points on the graph, and you will notice that as x increases, the y-values increase slowly. Draw a smooth curve passing through these points.
Now let's analyze the domain and range of each function based on the graph:
For the function y = 5^x, the domain is all real numbers (-∞, ∞) because any real number can be raised to any exponent. The range is all positive numbers (0, ∞) because the exponential function produces positive values.
For the function y = log_5 x, the domain is all positive numbers (0, ∞) because logarithms are only defined for positive values. The range is all real numbers (-∞, ∞) because the logarithmic function can take any real number as input.
Moving on to the y-intercept and asymptotes:
For y = 5^x, the y-intercept is (0, 1) since when x = 0, y = 5^0 = 1. There are no horizontal or vertical asymptotes.
For y = log_5 x, there is no y-intercept because the logarithm function is not defined for x = 0. The vertical asymptote is x = 0, which means the graph gets closer and closer to the y-axis as x approaches 0.
By following these steps, you can plot the graphs of y = 5^x and y = log_5 x, and use them to determine the domain, range, y-intercept, and any asymptotes for each function.