Find the inverse of the following function. Find the domain, range, and asymptotes of each function.
Graph both functions on the same coordinate plane
-x/3
f(x)=4+e
( if this makes sense, not sure how else to show it. the -x/3 is after the e like -x/3 to the power of e)
please help me! show steps also, if i an just given an answer that wont help me! thanks in advance!!!
y = 4+e^-x/3
y-4 = e^-x/3
ln(y-4) = -x/3
x = -3ln(y-4)
so, f inverse = -3ln(x-4)
for domain, recall that ln(u) is defined only for u>0
domain for exponentials and range for logs is all reals
To find the inverse of a function, we need to switch the roles of x and y and solve for y.
1. Inverse of -x/3:
Let's start with the function -x/3. To find its inverse, we interchange x and y:
x = -y/3
Next, we solve for y:
Multiply both sides of the equation by -3 to isolate y:
-3x = y
So, the inverse of -x/3 is y = -3x.
2. Inverse of 4+e:
Now, let's find the inverse of 4+e. Again, interchange x and y:
x = 4 + e
To solve for y, subtract 4 from both sides:
x - 4 = e
Now, the inverse of 4+e is y = x - 4.
Domain, Range, and Asymptotes:
To determine the domain and range of a function, we need to consider any restrictions on the variables.
For the function -x/3, there are no restrictions on the variable x. Thus, the domain is all real numbers. For the range, since -x/3 is a linear function, the range is also all real numbers.
For the function 4+e, there are no restrictions on the variable x. Therefore, the domain is all real numbers. For the range, since the graph of 4+e is a horizontal line parallel to the x-axis, the range is all real numbers.
As for asymptotes, both functions do not have any vertical or horizontal asymptotes because they are both linear functions.
Graph:
To graph the functions -x/3 and 4+e on the same coordinate plane, plot some points and connect them to form the lines.
For -x/3, choose different x-values, calculate the corresponding y-values, and plot the points. Connect the points to form a line.
For example, let's choose x = -3, -2, -1, 0, 1, 2, 3.
When x = -3, y = -(-3)/3 = 1. Plot (-3, 1).
When x = -2, y = -(-2)/3 = 2/3. Plot (-2, 2/3).
When x = -1, y = -(-1)/3 = 1/3. Plot (-1, 1/3).
When x = 0, y = -(0)/3 = 0. Plot (0, 0).
When x = 1, y = -(1)/3 = -1/3. Plot (1, -1/3).
When x = 2, y = -(2)/3 = -2/3. Plot (2, -2/3).
When x = 3, y = -(3)/3 = -1. Plot (3, -1).
Now, for 4+e, it is a horizontal line that intersects the y-axis at 4. So, plot the point (0, 4) and draw a horizontal line passing through that point.
Finally, label the axes, and you will have the graph of the functions on the same coordinate plane.