Find the inverse of f(x) = 2x - 5 and verify by composition.

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Multiple Choice

Find the inverse of f(x) = 2x - 5 and verify by composition.

Explanation:
When you find an inverse, you undo the function by swapping x and y and solve for y. For f(x) = 2x - 5, let y = 2x - 5. Swap: x = 2y - 5. Solve for y: 2y = x + 5, so y = (x + 5)/2. The inverse is f^{-1}(x) = (x + 5)/2. Verify by composition: f(f^{-1}(x)) = 2·[(x + 5)/2] - 5 = (x + 5) - 5 = x, so composing the functions gives x as expected. The other proposed inverses fail the composition check (for example, (x - 5)/2 gives f^{-1}(f(x)) = x - 5, and x/2 - 5 gives f(f^{-1}(x)) = x - 15), confirming the inverse above is correct.

When you find an inverse, you undo the function by swapping x and y and solve for y. For f(x) = 2x - 5, let y = 2x - 5. Swap: x = 2y - 5. Solve for y: 2y = x + 5, so y = (x + 5)/2. The inverse is f^{-1}(x) = (x + 5)/2.

Verify by composition: f(f^{-1}(x)) = 2·[(x + 5)/2] - 5 = (x + 5) - 5 = x, so composing the functions gives x as expected. The other proposed inverses fail the composition check (for example, (x - 5)/2 gives f^{-1}(f(x)) = x - 5, and x/2 - 5 gives f(f^{-1}(x)) = x - 15), confirming the inverse above is correct.

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