`y-b=a-2^(-x)`
For the equation given, if `agt0 and blt0.` then which of the following statements is always true?

Options

A. `ylta+b`
B. `ygta+b`
C. `y=a+b`
D. `y=-2(a+b)`

Explanation

To solve the equation \( y - b = a - 2^{-x} \) and determine which statement is always true given that \( a > 0 \) and \( b < 0 \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ y - b = a - 2^{-x} \]
Add \( b \) to both sides to isolate \( y \):
\[ y = a - 2^{-x} + b \]

### Step 2: Combine the constants
Now, combine the constants \( a \) and \( b \):
\[ y = (a + b) - 2^{-x} \]

### Step 3: Analyze the term \( 2^{-x} \)
The term \( 2^{-x} \) can be rewritten as \( \frac{1}{2^x} \). Since \( 2^x \) is always positive for any real \( x \), it follows that \( 2^{-x} \) is also always positive. Therefore, \( 2^{-x} > 0 \).

### Step 4: Determine the relationship between \( y \) and \( a + b \)
Since \( 2^{-x} > 0 \), we can conclude:
\[ y = (a + b) - 2^{-x} < a + b \]
This means that \( y \) is always less than \( a + b \).

### Conclusion
Thus, the statement that is always true is:
\[ y < a + b \]