Harsh made a piggy bank for himself using clay . If the shape of the bank is based on the function f (x)= | x - 4|+ | x - 5 | where f (x) represents the height of the bank

For ` x in (4, 5)` the value of the functions f (x) is

Options

A. ` 2 x - 9 `
B. 1
C. `-1`
D. 2 x + 1

Answer

To find the value of the function \( f(x) = |x - 4| + |x - 5| \) for \( x \) in the interval \( (4, 5) \), we will follow these steps:

### Step 1: Understand the Function
The function consists of two absolute value expressions: \( |x - 4| \) and \( |x - 5| \). We need to evaluate this function for values of \( x \) that lie strictly between 4 and 5.

**Hint:** Recall that the absolute value function \( |a| \) is defined as:
- \( a \) if \( a \geq 0 \)
- \( -a \) if \( a < 0 \)

### Step 2: Evaluate \( |x - 4| \)
Since \( x \) is in the interval \( (4, 5) \), we can determine that \( x - 4 \) is positive:
- \( |x - 4| = x - 4 \)

**Hint:** Since \( x \) is greater than 4, \( x - 4 \) will always be positive in this interval.

### Step 3: Evaluate \( |x - 5| \)
Now, we evaluate \( |x - 5| \). Since \( x \) is less than 5 in this interval, \( x - 5 \) will be negative:
- \( |x - 5| = -(x - 5) = 5 - x \)

**Hint:** Remember that because \( x < 5 \), the expression \( x - 5 \) is negative, and we need to take the negative of that.

### Step 4: Combine the Results
Now, substitute the results from Steps 2 and 3 into the function \( f(x) \):
\[ f(x) = |x - 4| + |x - 5| = (x - 4) + (5 - x) \]

### Step 5: Simplify the Expression
Now, simplify the expression:
\[ f(x) = x - 4 + 5 - x = 1 \]

### Conclusion
Thus, for \( x \) in the interval \( (4, 5) \), the value of the function \( f(x) \) is:
\[ f(x) = 1 \]

**Final Answer:** The value of \( f(x) \) for \( x \) in \( (4, 5) \) is 1.

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