**Inverse Functions Using Tables :**

Here we are going to see, how we find inverse functions using tables.

For functions whose domain consists of only finitely many numbers, tables provide good insight into the notion of an inverse function.

Suppose f is a function defined by a table. Then:

- f is one-to-one if and only if the table defining f has no repetitions in the second column.
- If f is one-to-one, then the table for f −1 is obtained by interchanging the columns of the table defining f.

Let us look into some example problems to understand the concept.

**Question 1 :**

Suppose f is the function whose domain is the four numbers {√2, 8, 17, 18}, with the values of f given in the table shown here in the margin.

(a) What is the range of f ?

(b) Explain why f is a one-to-one function.

(c) What is the table for the function f ^{−1}?

**Solution :**

(a) From the table x represents domain and f(x) represents range. Hence the range of the function is {3, -5, 6, 1}.

(b) From the table, we know that every element of x is associated with unique elements of f(x). Hence it is one to one function.

(c) From the table, we understood that

f(√2) = 3, f(8) = -5, f(17) = 6, f(18) = 1

f^{-1}(3) = √2, f^{-1}(-5) = 8, f^{-1}(6) = 17, f^{-1}(1) = 18

**Question 2 :**

For f and g are functions, each of whose domain consists of four numbers, with f and g defined by the tables below:

(a) What is the domain of f ?

(b) What is the range of f ?

(c) Give the table of values for f^{−1}.

(d) What is the domain of f^{−1}?

(e) What is the range of f^{−1}?

(f) Sketch the graph of f and f^{-1}

(g) Give the table of values for f^{−1} ◦ f

**Solution :**

(a) Domain of f = {1, 2, 3, 4}

(b) Range of f = {2, 3, 4, 5}

(c) Give the table of values for f^{−1}.

(d) domain of f^{-1} = {2, 3, 4, 5}

(e) Range of f^{-1} = {1, 2, 3, 4}

(f) From the graph, we know that the inverse function is reflection of the function.

(g) Give the table of values for f◦f^{−1}

(f◦f ^{−1})(2) = f (f^{−1}(2)) = f(3) = 2

(f◦f ^{−1})(3) = f(f^{−1}(3)) = f(4) = 3

(f◦f^{−1})(4) = f(f ^{−1}(4)) = f(1) = 4

(f◦f^{ −1})(5) = f(f^{−1}(5)) = f(2) = 5

After having gone through the stuff given above, we hope that the students would have understood "Inverse Functions Using Tables".

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