Question

I need help asap

1: Write a function that represents this process:
Take a number, x.
Multiply it by 4.
Subtract 2 from the result.
Then find the inverse of this function. Does the inverse represent the reverse of the process above? Explain why or why not.
2:Show why, for linear functions, a vertical translation is equivalent to a horizontal translation. For a linear function, what horizontal translation is equivalent to a vertical translation of 3 units up?
3:Alex says that the function f(x) = (3x)2 represents a vertical stretch of the quadratic parent function by a factor of 3. Marta says that it represents a horizontal compression by a factor of . Decide whether one student is correct, both are correct, or neither is correct.
4:For an unknown parent function f(x), write a function g(x) that is:
vertically stretched by a factor of 2,
shifted up 5 units, and
shifted right 4 units.
Explain how your function accomplishes these transformations.

Answer 1

To write the function that represents the given process and its inverse, we start with the equation f(x) = 4x - 2 and find the inverse by swapping x and f(x). A vertical translation is equivalent to a horizontal translation for linear functions. The function f(x) = (3x)2 represents a vertical stretch of the quadratic parent function, but not a horizontal compression. To vertically stretch by 2, shift up 5 units, and shift right 4 units, the function g(x) = 2 * f(x - 4) + 5 can be used.

Step-by-step explanation:

1. To write a function that represents the given process, we can start with the equation f(x) = 4x - 2. To find the inverse of this function, we swap the roles of x and f(x) and solve for x. That gives us the equation f-1(x) = (x + 2) / 4. The inverse function represents the reverse of the process because applying the original function followed by its inverse will give us the original input.
2. For linear functions, a vertical translation is equivalent to a horizontal translation. To see why, consider the equation y = mx + b. Adding a constant value to y shifts the graph vertically, while adding a constant value to x shifts the graph horizontally. So, a vertical translation of 3 units up is equivalent to a horizontal translation of 3 units to the right.
3. Neither Alex nor Marta is completely correct. The function f(x) = (3x)2 represents a vertical stretch of the quadratic parent function, but not by a factor of 3. It actually represents a vertical stretch by a factor of 9. However, it does not represent a horizontal compression.
4. To achieve the desired transformations, we can start with the parent function f(x), and apply the necessary steps. To vertically stretch by a factor of 2, we multiply the function by 2 (f(x) * 2). To shift it up 5 units, we add 5 to the function (f(x) + 5). And to shift it right 4 units, we replace x with (x - 4) in the function. So, the final function g(x) that accomplishes these transformations is g(x) = 2 * f(x - 4) + 5.