If `f : R rarr R, f(x) = x^(2) + 2x - 3` and `g : R rarr R, g(x) = 3x - 4` then the value of fog (x) is

Options

A. `3x^(2) + 6x - 13`
B. `9x^(2) - 18 x + 5`
C. `(3x - 4)^(2) + 2x - 3`
D. `x^(2) + 1`

Answer

To find the value of \( f(g(x)) \), we need to substitute \( g(x) \) into \( f(x) \). Let's go through the steps systematically.

### Step 1: Identify the functions
We have:
- \( f(x) = x^2 + 2x - 3 \)
- \( g(x) = 3x - 4 \)

### Step 2: Substitute \( g(x) \) into \( f(x) \)
To find \( f(g(x)) \), we replace \( x \) in \( f(x) \) with \( g(x) \):
\[ f(g(x)) = f(3x - 4) \]

### Step 3: Calculate \( f(3x - 4) \)
Now we will substitute \( 3x - 4 \) into the function \( f(x) \):
\[ f(3x - 4) = (3x - 4)^2 + 2(3x - 4) - 3 \]

### Step 4: Expand \( (3x - 4)^2 \)
Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ (3x - 4)^2 = (3x)^2 - 2 \cdot 3x \cdot 4 + 4^2 = 9x^2 - 24x + 16 \]

### Step 5: Expand \( 2(3x - 4) \)
Now calculate \( 2(3x - 4) \):
\[ 2(3x - 4) = 6x - 8 \]

### Step 6: Combine all parts
Now we combine all the parts together:
\[ f(3x - 4) = (9x^2 - 24x + 16) + (6x - 8) - 3 \]

### Step 7: Simplify the expression
Combine like terms:
\[ f(3x - 4) = 9x^2 - 24x + 6x + 16 - 8 - 3 \]
\[ = 9x^2 - 18x + 5 \]

### Final Answer
Thus, the value of \( f(g(x)) \) is:
\[ f(g(x)) = 9x^2 - 18x + 5 \]

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