To determine whether $f(x) = 2x + 5$ and $g(x) = \frac{x - 5}{2}$ are inverse functions, we need to check whether $f(g(x)) = x$ and $g(f(x)) = x$.

Step 1: Check $f(g(x))$

Substitute $g(x) = \frac{x - 5}{2}$ into $f(x) = 2x + 5$: $f(g(x)) = 2\left(\frac{x - 5}{2}\right) + 5$ Simplify: $f(g(x)) = (x - 5) + 5 = x$

Step 2: Check $g(f(x))$

Substitute $f(x) = 2x + 5$ into $g(x) = \frac{x - 5}{2}$: $g(f(x)) = \frac{(2x + 5) - 5}{2}$ Simplify: $g(f(x)) = \frac{2x}{2} = x$

Conclusion:

Since both $f(g(x)) = x$ and $g(f(x)) = x$, the functions $f(x) = 2x + 5$ and $g(x) = \frac{x - 5}{2}$ are inverse functions.

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Related Questions:

1. How do you find the inverse of a function algebraically?
2. What are the conditions for two functions to be inverses?
3. How do graphs of inverse functions relate to each other?
4. How does the domain of a function relate to the range of its inverse?
5. What is the importance of one-to-one functions in inverses?

Tip:

When verifying inverse functions, always confirm both $f(g(x)) = x$ and $g(f(x)) = x$ to avoid errors!