If `f(x)=|x| and g(x)=[x]` then value of fog `(-(1)/(4))+ gof (-(1)/(4))` is

Options

A. 0
B. 1
C. `-1`
D. `1//4`

Answer

To solve the problem, we need to find the value of \( f(g(-\frac{1}{4})) + g(f(-\frac{1}{4})) \) where \( f(x) = |x| \) and \( g(x) = [x] \) (the greatest integer function).

### Step 1: Calculate \( g(-\frac{1}{4}) \)
The function \( g(x) = [x] \) gives the greatest integer less than or equal to \( x \).

For \( x = -\frac{1}{4} \):
\[ g(-\frac{1}{4}) = [-\frac{1}{4}] = -1 \]

### Step 2: Calculate \( f(g(-\frac{1}{4})) \)
Now we need to find \( f(g(-\frac{1}{4})) = f(-1) \).

The function \( f(x) = |x| \) gives the absolute value of \( x \).

So,
\[ f(-1) = |-1| = 1 \]

### Step 3: Calculate \( f(-\frac{1}{4}) \)
Next, we need to find \( f(-\frac{1}{4}) \).

Using \( f(x) = |x| \):
\[ f(-\frac{1}{4}) = |-\frac{1}{4}| = \frac{1}{4} \]

### Step 4: Calculate \( g(f(-\frac{1}{4})) \)
Now we need to find \( g(f(-\frac{1}{4})) = g(\frac{1}{4}) \).

Using \( g(x) = [x] \):
\[ g(\frac{1}{4}) = [\frac{1}{4}] = 0 \]

### Step 5: Combine the results
Now we can combine the results from Step 2 and Step 4:
\[ f(g(-\frac{1}{4})) + g(f(-\frac{1}{4})) = 1 + 0 = 1 \]

### Final Answer:
The value of \( f(g(-\frac{1}{4})) + g(f(-\frac{1}{4})) \) is \( \boxed{1} \).

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