Let `f : R to R and g : R to R` be given by `f (x) = x^(2) and g(x) =x ^(3) +1,` then (fog) (x)

Options

A. `x ^(6)+1`
B. `x ^(6) -1`
C. `(x ^(3) -1)^(2)`
D. `(x ^(3) + 1)^(2)`

Answer

To find \((f \circ g)(x)\), we need to evaluate \(f(g(x))\). Let's break this down step by step.

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

### Step 2: Substitute \(g(x)\) into \(f(x)\)
We need to find \(f(g(x))\). This means we will replace \(x\) in \(f(x)\) with \(g(x)\):
\[ f(g(x)) = f(x^3 + 1) \]

### Step 3: Apply the function \(f\)
Now, we will apply the function \(f\) to \(g(x)\):
\[ f(x^3 + 1) = (x^3 + 1)^2 \]

### Step 4: Expand the expression
Next, we need to expand \((x^3 + 1)^2\):
\[ (x^3 + 1)^2 = x^6 + 2x^3 + 1 \]

### Final Result
Thus, we have:
\[ (f \circ g)(x) = x^6 + 2x^3 + 1 \]