How to Evaluate Functions at Specific Values (Step-by-Step Guide)

📚 Understanding Function Evaluation

Function evaluation is the process of finding the value of a function at a specific input. Imagine a function like a machine: you put something in (the input), the machine does something to it, and something comes out (the output). When we evaluate a function, we're figuring out what comes out for a particular input.

📜 A Brief History

The concept of functions has evolved over centuries. Early notions of functions were geometric, relating curves and shapes. Over time, mathematicians like Leibniz and Bernoulli contributed to formalizing the idea of a function as a relationship between variables. Euler introduced the notation $f(x)$, which is still widely used today. The rigorous definition of a function as a mapping between sets came later, solidifying its place in modern mathematics.

🔑 Key Principles of Function Evaluation

• 🔍 Function Notation: Understanding the notation $f(x)$. The letter $f$ represents the function, and $x$ is the input variable. $f(x)$ is the output (also called the value of the function at $x$).
• ➡️ Substitution: Replacing the input variable ($x$) with the given value. This is the core of function evaluation.
• 🔢 Order of Operations: Following the correct order of operations (PEMDAS/BODMAS) to simplify the expression after substitution.
• ➕ Careful Arithmetic: Avoiding simple arithmetic errors. Double-check your calculations!

✅ Step-by-Step Guide to Evaluating Functions

1. ✏️ Write down the function: Clearly identify the function you are working with. For example, $f(x) = 3x + 2$.
2. ➡️ Identify the input value: Determine the value you are substituting for $x$. For example, evaluate $f(x)$ at $x = 4$.
3. 🔄 Substitute: Replace every instance of $x$ in the function with the given value. In our example, $f(4) = 3(4) + 2$.
4. 🧮 Simplify: Perform the arithmetic operations according to the order of operations. $f(4) = 12 + 2 = 14$.
5. ✔️ State the answer: Clearly state the result. In this case, $f(4) = 14$.

💡 Real-World Examples

• 📈 Example 1: Linear Function

  Let $f(x) = 2x - 5$. Evaluate $f(3)$.

  Solution: $f(3) = 2(3) - 5 = 6 - 5 = 1$.

• 🌡️ Example 2: Quadratic Function

  Let $g(x) = x^2 + 3x - 1$. Evaluate $g(-2)$.

  Solution: $g(-2) = (-2)^2 + 3(-2) - 1 = 4 - 6 - 1 = -3$.
• ➗ Example 3: Rational Function

  Let $h(x) = \frac{x + 1}{x - 2}$. Evaluate $h(5)$.

  Solution: $h(5) = \frac{5 + 1}{5 - 2} = \frac{6}{3} = 2$.

• ➕ Example 4: Function with Multiple Variables

  Let $k(x, y) = x^2 + 2xy + y^2$. Evaluate $k(1, -1)$.

  Solution: $k(1, -1) = (1)^2 + 2(1)(-1) + (-1)^2 = 1 - 2 + 1 = 0$.

• ∛ Example 5: Function with Radicals

  Let $p(x) = \sqrt{x + 5}$. Evaluate $p(4)$.

  Solution: $p(4) = \sqrt{4 + 5} = \sqrt{9} = 3$.
• 🧊 Example 6: Piecewise Function

  Let $q(x) = \begin{cases} x + 1, & \text{if } x < 0 \\ x^2, & \text{if } x \geq 0 \end{cases}$. Evaluate $q(-3)$ and $q(2)$.

  Solution: Since $-3 < 0$, $q(-3) = -3 + 1 = -2$. Since $2 \geq 0$, $q(2) = 2^2 = 4$.

• 📐 Example 7: Trigonometric Function

  Let $r(x) = \sin(x)$. Evaluate $r(\frac{\pi}{2})$.

  Solution: $r(\frac{\pi}{2}) = \sin(\frac{\pi}{2}) = 1$.

📝 Practice Quiz

Evaluate the following functions at the given values:

1. $f(x) = 4x - 7$ at $x = 2$
2. $g(x) = x^2 - 5x + 6$ at $x = 1$
3. $h(x) = \frac{2x}{x + 3}$ at $x = 4$
4. $k(x) = \sqrt{3x + 1}$ at $x = 5$
5. $p(x) = x^3 - 2x + 1$ at $x = -1$
6. $q(x) = |x - 4|$ at $x = -2$
7. $r(x) = e^x$ at $x = 0$

🏁 Conclusion

Function evaluation is a fundamental skill in mathematics. By understanding the notation, following the order of operations, and practicing with different types of functions, you can master this essential concept. Remember to double-check your work and take your time – accuracy is key! Happy evaluating!