Question

Determine whether the function is one-to-one. $$ f(x)=\sqrt{1-x} $$

Short answer

The given function is \(f(x) = \sqrt{1-x}\). By using the definition of a one-to-one function and setting \(f(x_1) = f(x_2)\), we set up and solve the equation \(\sqrt{1-x_1} = \sqrt{1-x_2}\), which results in \(x_1 = x_2\). Since x1 and x2 are equal under the condition f(x1) = f(x2), the function is one-to-one.

Step-by-step solution

Set up using the definition of a one-to-one function.

Let f(x1) = f(x2), where x1 and x2 are any two points in the domain of the function. Given function: $$ f(x) = \sqrt{1 - x} $$ Set up the equation: $$ \sqrt{1 - x_{1}} = \sqrt{1 - x_{2}} $$

Solve for x1 and x2.

To solve for x1 and x2, we need to eliminate the square root on both sides. We can do this by squaring both sides of the equation: $$ (1 - x_{1}) = (1 - x_{2}) $$ Now, we can try to isolate x1 and x2. Subtracting (1 - x2) from both sides to obtain x1 in terms of x2: $$ x_{1} - x_{2} = 0 $$ $$ x_{1} = x_{2} $$ Since we find that x1 = x2 under the condition f(x1) = f(x2), we can conclude that the function is one-to-one.

Key concepts

Functions in Calculus

In calculus, a function represents a relationship between two sets, typically called the domain and the range. A function assigns exactly one element of the range to each element of the domain. In educational terms, you can think of it as a rule where you plug in a number (from the domain), and get a unique result (from the range).

When you come across functions like f(x) = \( \)sqrt{1-x}, it's not just about understanding the notation but also about how it behaves. Does it pass the Horizontal Line Test for being one-to-one, or another way of saying is that each value of 'f(x)' is unique for a different 'x'. This uniqueness of output values is crucial because it allows us to confidently reverse the function, finding the original input from a given output, which is the foundation of inverse functions — a topic essential in calculus.

Solving Equations

Solving equations is tantamount to cracking a code. You perform various operations to determine the unknown value that makes the equation true. In our case, we're not just solving for a numerical value but rather demonstrating a property of the function. You can visualize this as a detective inspecting two separate instances and trying to verify if they lead to the same conclusion.

The algebraic 'sleuthing' starts with setting two outputs of the function equal to one another, as in f(x1) = f(x2), then extracting the variables x1 and x2 from their mathematical 'disguises'. Here, squaring both sides is like 'picking a lock' to remove the square roots, allowing us to see if x1 and x2 refer to the same point. If, after all the manipulation, we end up with a statement where x1 = x2, we've 'cracked the code', confirming that the initial function indeed has the quality of being one-to-one.

Domain of a Function

The domain of a function is the set of all the inputs over which the function is defined. In simple terms, it's like the 'guest list' for a party — it specifies who's allowed to participate (which values can be substituted into 'x'). We must determine the domain first to understand the function's behavior and to avoid any 'uninvited guests' (like dividing by zero, or taking the square root of a negative number, which are no-nos in the realm of real numbers!).

For our function f(x) = \( \)sqrt{1-x}, the domain would include all 'x' values for which 1-x is greater than or equal to zero. Why? Because the square root function only works with non-negative numbers. It's this understanding of the domain that sets the stage for us to explore whether f(x) is one-to-one on its allowed interval. In conclusion, identifying the domain is like drawing the 'playing field' boundaries before the game begins.