Let functions `f,g:RtoR` defined by `f(x)=3x-1+abs(2x-1),g(x)=1/5(3x+5-abs(2x+5))`
Then. which relation between f and g is/are true

Options

A. gof= fog
B. `(fog)^(-1)=gof`
C. `(fof)=gog`
D. `(fof)^(-1)=gog`

Answer

To analyze the functions \( f \) and \( g \) and determine the relationship between them, we will first break down each function based on the definition provided.

### Step 1: Define the functions
The functions are defined as follows:
- \( f(x) = 3x - 1 + |2x - 1| \)
- \( g(x) = \frac{1}{5}(3x + 5 - |2x + 5|) \)

### Step 2: Analyze \( f(x) \)
We need to consider the absolute value in \( f(x) \):
- The expression \( |2x - 1| \) changes depending on the value of \( x \):
- If \( x < \frac{1}{2} \), then \( |2x - 1| = -(2x - 1) = -2x + 1 \)
- If \( x \geq \frac{1}{2} \), then \( |2x - 1| = 2x - 1 \)

Thus, we can write \( f(x) \) in piecewise form:
1. For \( x < \frac{1}{2} \):
\[ f(x) = 3x - 1 - 2x + 1 = x \]
2. For \( x \geq \frac{1}{2} \):
\[ f(x) = 3x - 1 + 2x - 1 = 5x - 2 \]

### Step 3: Analyze \( g(x) \)
Next, we analyze \( g(x) \):
- The expression \( |2x + 5| \) also changes based on \( x \):
- If \( x < -\frac{5}{2} \), then \( |2x + 5| = -(2x + 5) = -2x - 5 \)
- If \( x \geq -\frac{5}{2} \), then \( |2x + 5| = 2x + 5 \)

Thus, we can write \( g(x) \) in piecewise form:
1. For \( x < -\frac{5}{2} \):
\[ g(x) = \frac{1}{5}(3x + 5 + 2x + 5) = \frac{1}{5}(5x + 10) = x + 2 \]
2. For \( x \geq -\frac{5}{2} \):
\[ g(x) = \frac{1}{5}(3x + 5 - (2x + 5)) = \frac{1}{5}(x) = \frac{x}{5} \]

### Step 4: Compare \( f(x) \) and \( g(x) \)
Now we have:
- \( f(x) = \begin{cases} x & \text{if } x < \frac{1}{2} \\ 5x - 2 & \text{if } x \geq \frac{1}{2} \end{cases} \)

- \( g(x) = \begin{cases} x + 2 & \text{if } x < -\frac{5}{2} \\ \frac{x}{5} & \text{if } x \geq -\frac{5}{2} \end{cases} \)

### Step 5: Determine relationships
To find the relationship between \( f \) and \( g \), we can check specific values:
1. For \( x < -\frac{5}{2} \):
- \( f(x) \) is not defined for \( x < -\frac{5}{2} \) since \( f(x) \) is defined for all \( x \).
- \( g(x) = x + 2 \)

2. For \( -\frac{5}{2} \leq x < \frac{1}{2} \):
- \( g(x) = \frac{x}{5} \)
- \( f(x) = x \)

3. For \( x \geq \frac{1}{2} \):
- \( g(x) = \frac{x}{5} \)
- \( f(x) = 5x - 2 \)

### Conclusion
By substituting values and analyzing the piecewise functions, we can conclude that:
- \( f(x) \) and \( g(x) \) are not equal for all \( x \).
- They intersect at specific points, but generally, \( f(x) \) grows faster than \( g(x) \) for \( x \geq \frac{1}{2} \).