If the interval of difference is 3, then find ∆(2x - 5).

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Let's solve this step by step. We need to find the forward difference of the function 2x minus 5, given that the interval of difference is 3. The forward difference formula is delta f of x equals f of x plus h minus f of x, where h is the interval of difference. Now let's calculate f of x plus 3. We substitute x plus 3 into our function: f of x plus 3 equals 2 times the quantity x plus 3, minus 5. Expanding this gives us 2x plus 6 minus 5, which simplifies to 2x plus 1. Next, we apply the difference formula: delta of 2x minus 5 equals f of x plus 3 minus f of x, which is 2x plus 1 minus the quantity 2x minus 5. Now let's simplify the expression step by step. We have delta of 2x minus 5 equals 2x plus 1 minus the quantity 2x minus 5. Distributing the negative sign gives us 2x plus 1 minus 2x plus 5. Grouping like terms, we get 2x minus 2x plus 1 plus 5, which equals 0 plus 6, or simply 6. Notice that for a linear function, the forward difference is constant regardless of the x value, and it equals 6 for any interval of 3. To summarize what we've learned: The forward difference formula is delta f of x equals f of x plus h minus f of x. For linear functions, this difference is always constant. In our problem, delta of 2x minus 5 equals 6 when the interval is 3, and this result doesn't depend on the specific value of x.