The image contains two mathematical problems, each dealing with composition of functions.

Problem 1:

Functions:

• $f(x) = 2x - 1$
• $g(x) = x + 3$

You're tasked with finding $g(f(x))$, which is the composition of the functions $g \circ f$.

Steps:

1. $g(f(x)) = g(2x - 1)$
2. Substitute $2x - 1$ into $g(x)$: $g(2x - 1) = (2x - 1) + 3 = 2x + 2$

Then there are specific values for $g(f(-1))$ and $g(f(-5))$:

• $g(f(-1))$:

  1. $f(-1) = 2(-1) - 1 = -3$
  2. $g(-3) = -3 + 3 = 0$

• $g(f(-5))$:

  1. $f(-5) = 2(-5) - 1 = -11$
  2. $g(-11) = -11 + 3 = -8$

Problem 2:

Functions:

• $f(x) = x^2 + 14$
• $g(x) = 5x + 3$

You are again tasked with finding the composition of functions $g(f(x))$.

Steps:

1. $g(f(x)) = g(x^2 + 14)$
2. Substitute $x^2 + 14$ into $g(x)$: $g(x^2 + 14) = 5(x^2 + 14) + 3 = 5x^2 + 70 + 3 = 5x^2 + 73$

For specific values:

• $g(f(8))$:

  1. $f(8) = 8^2 + 14 = 64 + 14 = 78$
  2. $g(78) = 5(78) + 3 = 390 + 3 = 393$

• $g(f(6))$:

  1. $f(6) = 6^2 + 14 = 36 + 14 = 50$
  2. $g(50) = 5(50) + 3 = 250 + 3 = 253$

Would you like further clarification or more details on any of these steps?

Relative Questions:

1. How would you find $f(g(x))$ instead of $g(f(x))$?
2. What happens if you change the functions slightly, e.g., $f(x) = x^2 - 5x$ and $g(x) = 2x + 1$?
3. Can you find the derivative of the composition $g(f(x))$ if $f(x) = 3x^2$ and $g(x) = \sin(x)$?
4. What are the domain and range of $g(f(x))$ for these specific functions?
5. How would you solve for $x$ if you are given $g(f(x)) = 0$?

Tip:

When dealing with compositions, always start by substituting the inner function into the outer function carefully, and work step by step to avoid mistakes.