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Definition of the inverse of a function.
Let f and g be two functions such that
f(g(x)) = x for every x in the domain of g and
g(f(x)) = x for every x in the domain of f.
The function g is the inverse of the function f, and the domain of f is equal to the range of g, and vice versa.
Example: Find the inverse of y1 = 2x + 7
Solution
y1 = 2x + 7 interchange x and y;
x = 2y1 + 7 solve for y;
x - 7 = 2y1 + 7 -7 subtract 7 to both sides;
x - 7 = 2y1 divide by 2 both sides;
(x - 7)/2 = y1 replace y1 with y2;
y2 = (x - 7)/2
Thus, the inverse of y1 = 2x +7 is y2 = (x -7)/2
Let's check if this is true according to the above definition:
Let y1 = f(x) = 2x +7 and y2 = g(x) = (x -7)/2
1. f(g(x))= x ?
f(x) = 2x + 7
f((x - 7)/2) = 2[(x -7)/2] + 7 = x - 7 + 7 = x True
2. g(f(x) = x ?
g(x) = (x - 7)/2
g(2x + 7) = [(2x + 7) - 7]/2 = 2x/2 = x True
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What is the inverse function of add 6?
The inverse function means the opposite calculation. The inverse function of "add 6" would be "subtract 6".
If an inverse function undoes the work of the original function what does the original function's becomes the inverse function's domain?
Range
Does every function have an inverse that is a function?
No. A simple example of this is y = x2; the inverse is x = y2, which is not a function.
The inverse of the cosine function?
The inverse of the cosine is the secant.
What is an inverse of a function?
The opposite of another function - if you apply a function and then its inverse, you should get the original number back. For example, the inverse of squaring a positive number is taking the square root.
What is the relationships between inverse functions?
The inverse of the inverse is the original function, so that the product of the two functions is equivalent to the identity function on the appropriate domain. The domain of a function is the range of the inverse function. The range of a function is the domain of the inverse function.
Is the inverse of an exponential function the quadratic function?
No. The inverse of an exponential function is a logarithmic function.
If an inverse function undoes the work of the original function the original function's becomes the inverse function's domain?
The original function's RANGE becomes the inverse function's domain.
What is the inverse function of -6?
-6 is a number, not a function and so there is not an inverse function.
What is the inverse of a cubic function?
The inverse of the cubic function is the cube root function.
Why is x squared a inverse function?
X squared is not an inverse function; it is a quadratic function.
What is the inverse function of add 6?
The inverse function means the opposite calculation. The inverse function of "add 6" would be "subtract 6".
If an inverse function undoes the work of the original function what does the original function's becomes the inverse function's domain?
Range
Does every function have an inverse that is a function?
No. A simple example of this is y = x2; the inverse is x = y2, which is not a function.
If an inverse function undoes the work of the original function the original function's range becomes the inverse function's?
range TPate
The inverse of the cosine function?
The inverse of the cosine is the secant.
Is an exponential function is the inverse of a logarithmic function?
No, an function only contains a certain amount of vertices; leaving a logarithmic function to NOT be the inverse of an exponential function.
