Determine whether or not the given pairs of functions are inverses of each other. f(x)=0.75 x^2+

step 1 In this question, consider the following two functions f of x equal to 0 .75 x squared plus 2 an

Determine whether or not the given pairs of functions are...

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Key Concepts

Inverse Functions

Inverse functions are pairs of functions where the output of one function becomes the input of the other, effectively reversing the original transformation. If f and g are inverses, then composing them, i.e., f(g(x)) and g(f(x)), will return the original input x, provided that the operations are performed within the proper domains.

Function Composition

Function composition involves applying one function to the result of another. In the context of inverses, this means showing that applying f after g (and vice versa) yields the identity function on the appropriate set—this is a way to verify that the two functions truly undo each other’s operations.

Algebraic Verification of Inverses

To prove that two functions are inverses, one can perform algebraic manipulations where one function is substituted into the other and simplified. If the composition simplifies to x for all x in the relevant domain, then the functions are confirmed to be inverses.

Domain and Range Considerations

When dealing with functions involving operations like square roots or quadratics, special attention must be paid to the functions' domains and ranges. Compatibility between these is essential for the inverse relationship to hold, ensuring that the output of one function falls within the acceptable input values of the other.

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