Find the composition fog and gof where `f(x)=x^3+1,g(x)=x^2-2`

Answer

To find the compositions \( f \circ g \) and \( g \circ f \) for the functions \( f(x) = x^3 + 1 \) and \( g(x) = x^2 - 2 \), we will follow these steps:

### Step 1: Find \( f \circ g \)

To find \( f \circ g \), we need to substitute \( g(x) \) into \( f(x) \):

\[ f(g(x)) = f(x^2 - 2) \]

Now, we substitute \( x^2 - 2 \) into \( f(x) \):

\[ f(x^2 - 2) = (x^2 - 2)^3 + 1 \]

### Step 2: Expand \( (x^2 - 2)^3 \)

Using the binomial expansion formula \( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \):

Let \( a = x^2 \) and \( b = 2 \):

\[ (x^2 - 2)^3 = (x^2)^3 - 3(x^2)^2(2) + 3(x^2)(2^2) - 2^3 \]

Calculating each term:

- \( (x^2)^3 = x^6 \)
- \( -3(x^2)^2(2) = -6x^4 \)
- \( 3(x^2)(2^2) = 12x^2 \)
- \( -2^3 = -8 \)

Putting it all together:

\[ (x^2 - 2)^3 = x^6 - 6x^4 + 12x^2 - 8 \]

Now, adding 1:

\[ f(g(x)) = x^6 - 6x^4 + 12x^2 - 8 + 1 = x^6 - 6x^4 + 12x^2 - 7 \]

### Step 3: Find \( g \circ f \)

Now, we need to find \( g(f(x)) \):

\[ g(f(x)) = g(x^3 + 1) \]

Substituting \( x^3 + 1 \) into \( g(x) \):

\[ g(x^3 + 1) = (x^3 + 1)^2 - 2 \]

### Step 4: Expand \( (x^3 + 1)^2 \)

Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):

Let \( a = x^3 \) and \( b = 1 \):

\[ (x^3 + 1)^2 = (x^3)^2 + 2(x^3)(1) + 1^2 = x^6 + 2x^3 + 1 \]

Now, subtracting 2:

\[ g(f(x)) = x^6 + 2x^3 + 1 - 2 = x^6 + 2x^3 - 1 \]

### Final Results

Thus, the compositions are:

\[ f \circ g = x^6 - 6x^4 + 12x^2 - 7 \]
\[ g \circ f = x^6 + 2x^3 - 1 \]