Question:
Which of the following functions satisfies these two criteria: \( f(0) = 0 \) and \( f(x+1) = 2f(x) + 1 \)?
Which of the following functions satisfies these two criteria: \( f(0) = 0 \) and \( f(x+1) = 2f(x) + 1 \)?
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To solve these problems, carefully check both conditions before proceeding with the formula of your choice.
Updated On: Apr 19, 2025
- \( f(x) = 1 - 2^x \)
- \( f(x) = 2x - 1 \)
- \( f(x) = -(2x + 1) \)
- \( f(x) = 2x + 1 \)
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The Correct Option is D
Solution and Explanation
We are given the two conditions:
1. \( f(0) = 0 \)
2. \( f(x+1) = 2f(x) + 1 \)
Let's test the given options one by one.
For \( f(x) = 2x + 1 \), check the two conditions:
1. \( f(0) = 2(0) + 1 = 1 \neq 0 \) — This doesn't satisfy the condition \( f(0) = 0 \).
Checking option (2): \( f(x) = 2x - 1 \):
1. \( f(0) = 2(0) - 1 = -1 \neq 0 \) — Again, this doesn't satisfy the condition \( f(0) = 0 \).
After checking, none of the options satisfies the condition \( f(0) = 0 \) for the given question. Thus, this solution isn't correct.
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