Let f:N→ Y be a function defined as f(x)=4x+3, where, Y={y∈N:y=4x+3 for some x∈N}. Show that f is invertible. Find the inverse.
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Step-by-Step Solution
Step 1: Prove Injectivity (One-to-One)
To prove that a function is invertible, we first need to show it is injective (one-to-one). We assume that f(x1)=f(x2) for two elements x1,x2 in the domain N. By substituting the function definition, we get 4x1+3=4x2+3. Simplifying this equation leads to x1=x2, which confirms that f is injective.
Step 2: Prove Surjectivity (Onto)
Next, we need to prove that the function is surjective (onto). This means that for every element y in the codomain Y, there exists an element x in the domain N such that f(x)=y. We set y=f(x) and solve for x. The codomain Y is defined as all natural numbers y such that y=4x+3 for some x∈N. This implies that for any y∈Y, y−3 must be a multiple of 4, and y−3≥0. Since x∈N, x=4y−3 will always be a natural number for any y∈Y. Thus, f is surjective.
Step 3: Conclusion of Invertibility
Since the function f has been proven to be both injective (one-to-one) and surjective (onto), it satisfies the conditions for invertibility. Therefore, f is an invertible function.
Step 4: Find the Inverse Function
To find the inverse function, we use the expression for x that we derived when proving surjectivity. If y=f(x), then x=f−1(y). So, the inverse function f−1 maps an element y from the codomain Y back to its corresponding element x in the domain N.