Test Questions for f(g(x)) and g(f(x)) Composition: Algebra 2 Prep

📚 Quick Study Guide

• 🔍 Function Composition Basics: Function composition means plugging one function into another. We write it as $f(g(x))$, which means we first apply the function $g$ to $x$, and then apply the function $f$ to the result.
• 💡 Notation: $f(g(x))$ is read as "f of g of x." Remember to work from the inside out! Evaluate $g(x)$ first.
• 📝 Finding $f(g(x))$: To find $f(g(x))$, replace every instance of 'x' in the function $f(x)$ with the entire function $g(x)$.
• 🧮 Finding $g(f(x))$: Similarly, to find $g(f(x))$, replace every instance of 'x' in the function $g(x)$ with the entire function $f(x)$.
• ➕ Simplifying: After substituting, simplify the expression using algebraic techniques like distribution, combining like terms, and factoring.
• ➗ Domain Considerations: Be mindful of the domains of the functions. The domain of the composite function is restricted by the domains of both the inner and outer functions.
• ✅ Key Tip: Practice, practice, practice! The more you work with function compositions, the easier they become.

🧪 Practice Quiz

1. If $f(x) = 2x + 3$ and $g(x) = x^2$, what is $f(g(x))$?
   1. $2x^2 + 3$
   2. $4x^2 + 12x + 9$
   3. $x^4$
   4. $2x + 3x^2$
2. If $f(x) = x - 1$ and $g(x) = \sqrt{x}$, what is $g(f(x))$?
   1. $\sqrt{x} - 1$
   2. $\sqrt{x-1}$
   3. $\sqrt{x} - \sqrt{1}$
   4. $x - \sqrt{1}$
3. If $f(x) = x^2 + 1$ and $g(x) = 3x$, what is $f(g(2))$?
   1. 13
   2. 37
   3. 82
   4. 19
4. If $f(x) = \frac{1}{x}$ and $g(x) = x + 2$, what is $f(g(x))$?
   1. $\frac{1}{x+2}$
   2. $\frac{1}{x} + 2$
   3. $\frac{x}{1+2x}$
   4. $x + \frac{1}{2}$
5. If $f(x) = 4x - 5$ and $g(x) = \frac{x+5}{4}$, what is $g(f(x))$?
   1. $x$
   2. $\frac{16x - 20 + 5}{4}$
   3. $\frac{x}{4}$
   4. $16x^2 - 25$
6. If $f(x) = |x|$ and $g(x) = x - 3$, what is $f(g(x))$?
   1. $|x| - 3$
   2. $|x-3|$
   3. $x - |3|$
   4. $|x-3|^2$
7. If $f(x) = 5$ and $g(x) = x^2 - 2x + 1$, what is $f(g(x))$?
   1. $5x^2 - 10x + 5$
   2. $x^2 - 2x + 1$
   3. 5
   4. 0