High school calculus: Function operations and composition diagnostic test

📚 Quick Study Guide

• ➕ Addition: $(f+g)(x) = f(x) + g(x)$
• ➖ Subtraction: $(f-g)(x) = f(x) - g(x)$
• ✖️ Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$
• ➗ Division: $(f/g)(x) = f(x) / g(x)$, where $g(x) \neq 0$
• 🎯 Composition: $(f \circ g)(x) = f(g(x))$
• 💡 Domain of Composition: The domain of $f(g(x))$ is all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.
• 📝 Important Note: Pay attention to the domain of each function when performing operations!

Practice Quiz

1. What is $(f+g)(x)$ if $f(x) = x^2$ and $g(x) = 2x + 1$?
   1. $x^2 + 2x + 1$
   2. $x^2 - 2x - 1$
   3. $2x^3 + x^2$
   4. $x^2 + 2x$
2. What is $(f-g)(x)$ if $f(x) = 3x^2$ and $g(x) = x - 2$?
   1. $3x^2 - x - 2$
   2. $3x^2 - x + 2$
   3. $2x^2 + 2$
   4. $3x^2 + x - 2$
3. What is $(f \cdot g)(x)$ if $f(x) = x + 1$ and $g(x) = x - 1$?
   1. $x^2 + 1$
   2. $x^2 - 1$
   3. $x^2 + 2x + 1$
   4. $x^2 - 2x + 1$
4. What is $(f/g)(x)$ if $f(x) = x^2 - 4$ and $g(x) = x + 2$?
   1. $x + 2$
   2. $x - 2$
   3. $x^2 - 2$
   4. $1$
5. What is $(f \circ g)(x)$ if $f(x) = x^2$ and $g(x) = x + 3$?
   1. $x^2 + 3$
   2. $x^2 + 6x + 9$
   3. $x^2 + 9$
   4. $x + 3^2$
6. What is $(g \circ f)(x)$ if $f(x) = 2x$ and $g(x) = x - 1$?
   1. $2x - 1$
   2. $2x + 1$
   3. $2(x - 1)$
   4. $x - 2$
7. If $f(x) = \sqrt{x}$ and $g(x) = x - 4$, what is the domain of $f(g(x))$?
   1. $x \geq 0$
   2. $x \geq 4$
   3. $x > 4$
   4. All real numbers