show the inverse function of constructible of values for each of the following function
1.f(x)=x + 7 / 3 2. f(x)= x -2 / 5
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Answers
Answer:
We use the symbol f − 1 to denote an inverse function. For example, if f (x) and g (x) are inverses of each other, then we can symbolically represent this statement as:
g(x) = f − 1(x) or f(x) = g−1(x)
One thing to note about the inverse function is that the inverse of a function is not the same as its reciprocal, i.e., f – 1 (x) ≠ 1/ f(x). This article will discuss how to find the inverse of a function.
Since not all functions have an inverse, it is therefore important to check whether a function has an inverse before embarking on determining its inverse.
We check whether or not a function has an inverse in order to avoid wasting time trying to find something that does not exist.
One-to-one functions
So how do we prove that a given function has an inverse? Functions that have inverse are called one-to-one functions.
A function is said to be one-to-one if, for each number y in the range of f, there is exactly one number x in the domain of f such that f (x) = y.
In other words, the domain and range of one-to-one function have the following relations:
Domain of f−1 = Range of f.
Range of f−1 = Domain of f.
For example, to check if f(x) = 3x + 5 is one to one function given, f(a) = 3a + 5 and f(b) = 3b + 5.
⟹ 3a + 5 = 3b + 5
⟹ 3a = 3b
⟹ a = b.
Therefore, f (x) is one-to-one function because, a = b.
Consider another case where a function f is given by f = {(7, 3), (8, –5), (–2, 11), (–6, 4)}. This function is one-to-one because none of its y - values appear more than once.
What about this other function h = {(–3, 8), (–11, –9), (5, 4), (6, –9)}? Function h is not one-to-one because the y- value of –9 appears more than once.
You can also graphically check one-to-one function by drawing a vertical line and horizontal line through a function graph. A function is one-to-one if both the horizontal and vertical line passes through the graph once.