32. Show that f: R -> R defined as f(x) = x / sqrt(1 + x2) is one-one but not onto. (function, one-one, onto, real numbers, domain, codomain, range)
Get the complete, step-by-step math solution for: "32. Show that f: R -> R defined as f(x) = x / sqrt(1 + x²) is one-one but not onto. (function, one-one, onto, real numbers, domain, codomain, range)". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Define the function
We are given the function f(x) mapping real numbers to real numbers. We need to demonstrate that this function is one-one (injective) but not onto (surjective).
Step 2: Check for one-one property
To prove that a function is one-one, we assume that f(x1)=f(x2) for two elements x1,x2 in the domain. If this assumption leads to x1=x2, then the function is one-one.
Step 3: Set up the equality for one-one proof
Substitute the function definition into the assumption f(x1)=f(x2). We need to determine if this equation implies x1=x2. Remember that the sign of 1+x2 is always positive.
Step 4: Square both sides
To eliminate the square roots, we square both sides of the equation. This simplifies the expression and allows us to manipulate it further algebraically.
Step 5: Rearrange and simplify
Cross-multiply and expand the terms. Notice that the term x12x22 appears on both sides, cancelling out. This leaves us with x12=x22.
Step 6: Consider square root carefully
From x12=x22, it implies that x1=±x2. We need to show that x1 and x2 must have the same sign for f(x1)=f(x2) to hold.
Step 7: Conclude one-one property
If x1=−x2 (and x1=0), then f(x1)=1+x12x1 and f(x2)=1+(−x1)2−x1=1+x12−x1=−f(x1). If f(x1)=f(x2), then f(x1)=−f(x1), which means 2f(x1)=0, so f(x1)=0. This implies x1=0. Therefore, if f(x1)=f(x2), we must have x1=x2. Hence, the function is one-one.
Step 8: Check for onto property
To check if the function is onto, we need to find the range of f(x) and determine if it equals the codomain, which is R (all real numbers). We set y=f(x) and try to express x in terms of y.
Step 9: Solve for x in terms of y
Square both sides to eliminate the square root. Then, rearrange the equation to isolate x2. This allows us to determine the possible values of y for which x can be a real number.
Step 10: Determine the range of y
For x to be a real number, x2 must be non-negative. This means 1−y2y2 must be greater than or equal to 0. Since y2≥0, we require 1−y2>0. This inequality implies that −1<y<1 for all values in the range.
Step 11: Conclude onto property
The range of the function is the interval (−1,1), which means that for any real number y outside this interval (e.g., y=2), there is no real x such that f(x)=y. Since the range is not equal to the codomain R, the function is not onto.