To solve for $(f \circ g)(x)$ and $(g \circ f)(x)$, we need to recall the definition of composition of functions. For two functions $f(x)$ and $g(x)$:

• $(f \circ g)(x) = f(g(x))$, meaning you substitute $g(x)$ into $f(x)$.
• $(g \circ f)(x) = g(f(x))$, meaning you substitute $f(x)$ into $g(x)$.

Problem 11:

Given:

• $f(x) = 2x + 1$
• $g(x) = x - 3$

1. Find $(f \circ g)(x) = f(g(x))$:

$f(g(x)) = f(x - 3) = 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5$

2. Find $(g \circ f)(x) = g(f(x))$:

$g(f(x)) = g(2x + 1) = (2x + 1) - 3 = 2x - 2$

Problem 12:

Given:

• $f(x) = x^2 + 3$
• $g(x) = 2x - 1$

1. Find $(f \circ g)(x) = f(g(x))$:

$f(g(x)) = f(2x - 1) = (2x - 1)^2 + 3 = 4x^2 - 4x + 1 + 3 = 4x^2 - 4x + 4$

2. Find $(g \circ f)(x) = g(f(x))$:

$g(f(x)) = g(x^2 + 3) = 2(x^2 + 3) - 1 = 2x^2 + 6 - 1 = 2x^2 + 5$

Problem 13:

Given:

• $f(x) = -x^2 - 8$
• $g(x) = x^2 - 1$

1. Find $(f \circ g)(x) = f(g(x))$:

$f(g(x)) = f(x^2 - 1) = -(x^2 - 1)^2 - 8 = -(x^4 - 2x^2 + 1) - 8 = -x^4 + 2x^2 - 1 - 8 = -x^4 + 2x^2 - 9$

2. Find $(g \circ f)(x) = g(f(x))$:

$g(f(x)) = g(-x^2 - 8) = (-x^2 - 8)^2 - 1 = (x^4 + 16x^2 + 64) - 1 = x^4 + 16x^2 + 63$

Problem 14:

Given:

• $f(x) = x + 2$
• $g(x) = x - 2$

1. Find $(f \circ g)(x) = f(g(x))$:

$f(g(x)) = f(x - 2) = (x - 2) + 2 = x$

2. Find $(g \circ f)(x) = g(f(x))$:

$g(f(x)) = g(x + 2) = (x + 2) - 2 = x$

Summary:

• Problem 11:
  • $(f \circ g)(x) = 2x - 5$
  • $(g \circ f)(x) = 2x - 2$
• Problem 12:
  • $(f \circ g)(x) = 4x^2 - 4x + 4$
  • $(g \circ f)(x) = 2x^2 + 5$
• Problem 13:
  • $(f \circ g)(x) = -x^4 + 2x^2 - 9$
  • $(g \circ f)(x) = x^4 + 16x^2 + 63$
• Problem 14:
  • $(f \circ g)(x) = x$
  • $(g \circ f)(x) = x$

Would you like further explanation on any of these steps?

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Here are five related questions to test your understanding:

1. What is the composition $(f \circ g)(x)$ if $f(x) = 3x + 1$ and $g(x) = x^2 - 4$?
2. How would the result change for $(g \circ f)(x)$ if $f(x) = x^2 + 4$ and $g(x) = 2x + 1$?
3. Can the composition of two functions ever result in a constant value for all $x$?
4. What is the difference between $f(g(x))$ and $g(f(x))$ in terms of domain restrictions?
5. If $f(x) = x^3$ and $g(x) = \frac{1}{x}$, what is $(f \circ g)(x)$?

Tip: Always ensure that the range of the inner function is within the domain of the outer function when composing functions.