If `f,g: RvecR` are defined respectively by `f(x)=x^2+3x+1,g(x)=2x-3,` find fog (ii) gof (iii) fof (iv) gog.

Explanation

Let's solve the problem step by step.

### Given Functions:
1. \( f(x) = x^2 + 3x + 1 \)
2. \( g(x) = 2x - 3 \)

### (i) Find \( f(g(x)) \) or \( f \circ g \):
To find \( f(g(x)) \), we will substitute \( g(x) \) into \( f(x) \).

1. Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(2x - 3) = (2x - 3)^2 + 3(2x - 3) + 1 \]

2. Expand \( (2x - 3)^2 \):
\[ (2x - 3)^2 = 4x^2 - 12x + 9 \]

3. Expand \( 3(2x - 3) \):
\[ 3(2x - 3) = 6x - 9 \]

4. Combine all parts:
\[ f(g(x)) = 4x^2 - 12x + 9 + 6x - 9 + 1 \]
\[ = 4x^2 - 6x + 1 \]

### (ii) Find \( g(f(x)) \) or \( g \circ f \):
To find \( g(f(x)) \), we will substitute \( f(x) \) into \( g(x) \).

1. Substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(x^2 + 3x + 1) = 2(x^2 + 3x + 1) - 3 \]

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

### (iii) Find \( f(f(x)) \) or \( f \circ f \):
To find \( f(f(x)) \), we will substitute \( f(x) \) into itself.

1. Substitute \( f(x) \) into \( f(x) \):
\[ f(f(x)) = f(x^2 + 3x + 1) = (x^2 + 3x + 1)^2 + 3(x^2 + 3x + 1) + 1 \]

2. Expand \( (x^2 + 3x + 1)^2 \):
\[ = x^4 + 6x^3 + 9x^2 + 6x + 1 \]

3. Expand \( 3(x^2 + 3x + 1) \):
\[ = 3x^2 + 9x + 3 \]

4. Combine all parts:
\[ f(f(x)) = x^4 + 6x^3 + 9x^2 + 6x + 1 + 3x^2 + 9x + 3 + 1 \]
\[ = x^4 + 6x^3 + 12x^2 + 15x + 5 \]

### (iv) Find \( g(g(x)) \) or \( g \circ g \):
To find \( g(g(x)) \), we will substitute \( g(x) \) into itself.

1. Substitute \( g(x) \) into \( g(x) \):
\[ g(g(x)) = g(2x - 3) = 2(2x - 3) - 3 \]

2. Expand:
\[ = 4x - 6 - 3 \]
\[ = 4x - 9 \]

### Final Answers:
1. \( f(g(x)) = 4x^2 - 6x + 1 \)
2. \( g(f(x)) = 2x^2 + 6x - 1 \)
3. \( f(f(x)) = x^4 + 6x^3 + 12x^2 + 15x + 5 \)
4. \( g(g(x)) = 4x - 9 \)