(a) What is the inverse of the function y = 3x2 ?
(b) On the same set of axes, without the use of a graphical calculator, graph the original
function and its inverse.
(c) Is the inverse of the function also a function? Explain your decision.
a. Y = 3x^2.
X - 3y^2.
y^2 = x/3.
Y = sqrt(x/3).
(a) To find the inverse of the function y = 3x^2, we need to swap the x and y variables and solve for y. Let's start with the original equation:
y = 3x^2
Step 1: Swap x and y:
x = 3y^2
Step 2: Solve for y:
Divide both sides of the equation by 3:
x/3 = y^2
Step 3: Take the square root of both sides:
√(x/3) = y
So, the inverse of the function y = 3x^2 is y = √(x/3).
(b) To graph the original function and its inverse on the same set of axes without a graphical calculator, we can follow these steps:
Step 1: Choose a set of x-values that spans the range of interest.
Let's select x-values ranging from -3 to 3.
Step 2: Calculate the corresponding y-values for the original function.
Plug each x-value into y = 3x^2 to find the y-values.
For example:
When x = -3, y = 3(-3)^2 = 27
When x = -2, y = 3(-2)^2 = 12
When x = -1, y = 3(-1)^2 = 3
When x = 0, y = 3(0)^2 = 0
When x = 1, y = 3(1)^2 = 3
When x = 2, y = 3(2)^2 = 12
When x = 3, y = 3(3)^2 = 27
Step 3: Calculate the corresponding y-values for the inverse function.
Plug each x-value into y = √(x/3) to find the y-values.
For example:
When x = -3, y = √((-3)/3) = -1
When x = -2, y = √((-2)/3) = -0.82 (rounded to 2 decimal places)
When x = -1, y = √((-1)/3) = -0.58 (rounded to 2 decimal places)
When x = 0, y = √((0)/3) = 0
When x = 1, y = √((1)/3) = 0.58 (rounded to 2 decimal places)
When x = 2, y = √((2)/3) = 0.82 (rounded to 2 decimal places)
When x = 3, y = √((3)/3) = 1
Step 4: Plot the points and draw the graphs.
Plot the points (x,y) for the original function and the points (x,y) for its inverse. Then, connect the points to form the graphs of both functions.
(c) The inverse of a function is also a function if each input of the original function corresponds to exactly one output of its inverse, and vice versa.
In this case, since we found the inverse function y = √(x/3), it passes the horizontal line test. This means that for every x-value in the domain, there is only one corresponding y-value. Therefore, the inverse of the function y = 3x^2 is also a function.