What is Function Evaluation? (Grade 8 Math Explained)

📚 What is Function Evaluation?

Function evaluation is the process of finding the value of a function at a specific input. In simpler terms, it's like substituting a given number into a function and calculating the result. This result is the value of the function for that particular input.

📜 History and Background

The concept of functions has been around for centuries, with early ideas appearing in the work of mathematicians like Nicole Oresme in the 14th century. However, the formal notation and systematic study of functions, including evaluation, developed more fully in the 17th and 18th centuries with mathematicians like Leibniz, the Bernoullis, and Euler. Euler, in particular, formalized much of the notation we use today, such as $f(x)$ to represent a function.

🔑 Key Principles of Function Evaluation

• ➡️ Function Notation: Understanding that $f(x)$ means "the value of the function $f$ at $x$".
• 🔢 Substitution: Replacing the variable $x$ with the given value in the function's expression.
• 🧮 Order of Operations: Following the correct order of operations (PEMDAS/BODMAS) to calculate the result accurately.
• ✅ Simplification: Simplifying the expression after substitution to obtain the final value.

🌍 Real-World Examples

Let's look at a few examples to understand how function evaluation works:

1. Example 1: Given the function $f(x) = 3x + 2$, evaluate $f(4)$.
   • Substitute $x$ with $4$: $f(4) = 3(4) + 2$
   • Simplify: $f(4) = 12 + 2 = 14$
   • Therefore, $f(4) = 14$
2. Example 2: Given the function $g(x) = x^2 - 5$, evaluate $g(-2)$.
   • Substitute $x$ with $-2$: $g(-2) = (-2)^2 - 5$
   • Simplify: $g(-2) = 4 - 5 = -1$
   • Therefore, $g(-2) = -1$
3. Example 3: Given the function $h(x) = \frac{x + 3}{2}$, evaluate $h(7)$.
   • Substitute $x$ with $7$: $h(7) = \frac{7 + 3}{2}$
   • Simplify: $h(7) = \frac{10}{2} = 5$
   • Therefore, $h(7) = 5$

✍️ Practice Quiz

Evaluate the following functions:

1. If $f(x) = 2x - 1$, find $f(5)$.
2. If $g(x) = x^2 + 3x - 4$, find $g(-1)$.
3. If $h(x) = \frac{4x}{x + 2}$, find $h(3)$.
4. If $p(x) = -x + 7$, find $p(0)$.
5. If $q(x) = 2(x - 3)$, find $q(8)$.
6. If $r(x) = x^3 - 2x$, find $r(2)$.
7. If $s(x) = \frac{5 - x}{3}$, find $s(-4)$.

💡 Solutions to the Practice Quiz

1. $f(5) = 2(5) - 1 = 10 - 1 = 9$
2. $g(-1) = (-1)^2 + 3(-1) - 4 = 1 - 3 - 4 = -6$
3. $h(3) = \frac{4(3)}{3 + 2} = \frac{12}{5}$
4. $p(0) = -0 + 7 = 7$
5. $q(8) = 2(8 - 3) = 2(5) = 10$
6. $r(2) = (2)^3 - 2(2) = 8 - 4 = 4$
7. $s(-4) = \frac{5 - (-4)}{3} = \frac{5 + 4}{3} = \frac{9}{3} = 3$

🎯 Conclusion

Function evaluation is a fundamental skill in mathematics. By understanding the principles and practicing with examples, you can master this concept and build a strong foundation for more advanced topics. Keep practicing, and you'll become a pro in no time!