original function
[original function|original function] The original function is defined as a function with the property that its inverse function exists.
Definition
original function The original function is defined as a function with the property that its inverse function exists. This inverse function is called the inverse function of the original function. The original function contains a factor of 3x', while the du term is off by a constant multiplier of 6x*.
Mechanism
original function The original function contains a factor of 3x', not 6x*. This discrepancy results in du being off by a constant multiplier. Substituting x = 8 into the original function yields y = 4. The slope of the tangent line to the graph at x = 8 is determined by this calculation. The relationship between the original function and the tangent line is influenced by the constant multiplier and the factor of 3x' in the function.
Effects
original function The new function f~! reverses the original function f's effects by undoing its actions. This reversal directly impacts the outcomes previously established by f. The function f~! specifically targets and neutralizes the changes introduced by f. As a result, the original function's operations are effectively nullified through this counteraction.
Examples
original function Since the original function includes one factor of x* and du = 6x? dx, multiply both sides of the du equation by 1/6. This process ensures the equation remains balanced while adjusting for the substitution. The factor of 1/6 accounts for the scaling introduced by the derivative of u.
Constant Multiplier
original function The constant multiplier refers to a factor that scales the original function by a fixed value. In the given context, the original function contains a factor of 3x', indicating a constant multiplier rather than 6x*. This distinction highlights how the multiplier remains consistent across the function's application.
Effects on New Function
original function The new function f~! directly reverses the original function f by undoing its effects. This reversal is explicitly stated in the evidence, where f~! is described as undid what f did. The relationship between the two functions is defined by this inverse action, with f~! serving as the counterpart to f. The evidence highlights that the new function's primary purpose is to negate the original function's impact.
Since Dy Mechanism
original function The since dy mechanism involves calculating the slope of the tangent line to the graph at a specific x-value. When x = 8, substituting this value into the original function yields y = 4. This process demonstrates how the slope of the tangent line is determined by evaluating the function at the given x-coordinate. The result, y = 4, confirms the point of tangency on the graph.