Determine algebraically whether the function is one-to-one. Verify your answer graphically. If t

step 1 This is the function f of x is equal to under root x minus 2. We have to prove that this functio

Determine algebraically whether the function is one-to-one....

Key Concepts

Domain and Range Considerations

For functions involving operations like square roots, the domain (the set of permissible inputs) is restricted to values that make the expression inside the square root non-negative. These restrictions are important when determining both the function's behavior and its inverse, as they affect the allowable values for inputs and outputs.

Graphical Verification (Horizontal Line Test)

Graphical verification of a function’s one-to-one property can be accomplished using the horizontal line test. If no horizontal line intersects the graph of the function more than once, the function is one-to-one. This visual method complements the algebraic approach and provides additional confirmation of the function's injectivity.

Inverse Function

The inverse function of a one-to-one function reverses the roles of inputs and outputs. Finding an inverse typically involves interchanging the dependent and independent variables and then solving for the new dependent variable. The existence of an inverse is dependent on the one-to-one nature of the original function.

One-to-One Function

A one-to-one function, also known as an injective function, assigns distinct outputs to distinct inputs. This property is crucial because it ensures that each value in the range corresponds to exactly one value in the domain, which is a necessary condition for an inverse function to exist.

Injectivity (Algebraic Verification)

The injectivity of a function can be verified algebraically by showing that if f(a) equals f(b), then a must equal b. This method involves setting up the equation f(a) = f(b) and solving for the variables, which confirms that no two different inputs produce the same output.

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