Evaluate The Function At The Given Value.Given: $ H(x) = \frac{1}{3} \cdot 6^x $What Is $ H(2) $?

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Introduction

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In mathematics, evaluating a function at a given value is a fundamental concept that helps us understand the behavior of the function. It involves substituting the given value into the function and simplifying the expression to obtain the result. In this article, we will evaluate the function $h(x) = \frac{1}{3} \cdot 6^x$ at the given value $x = 2$.

Understanding the Function

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The given function is $h(x) = \frac{1}{3} \cdot 6^x$. This is an exponential function, where the base is 6 and the exponent is $x$. The function is multiplied by $\frac{1}{3}$, which is a constant factor. To evaluate the function at a given value, we need to substitute the value into the function and simplify the expression.

Evaluating the Function at x = 2

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To evaluate the function at $x = 2$, we substitute $x = 2$ into the function:

$h(2) = \frac{1}{3} \cdot 6^2$

Simplifying the Expression

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To simplify the expression, we need to evaluate the exponent first. The exponent is $6^2$, which is equal to $36$. Therefore, the expression becomes:

$h(2) = \frac{1}{3} \cdot 36$

Evaluating the Constant Factor

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The constant factor is $\frac{1}{3}$, which is a simple fraction. To evaluate the constant factor, we multiply it by $36$:

$h(2) = \frac{1}{3} \cdot 36 = 12$

Conclusion

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In conclusion, the value of the function $h(x) = \frac{1}{3} \cdot 6^x$ at $x = 2$ is $12$. This is obtained by substituting $x = 2$ into the function and simplifying the expression.

Example Use Cases

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Evaluating a function at a given value has many practical applications in mathematics and science. Some example use cases include:

• Physics: Evaluating the position of an object at a given time using the equation of motion.
• Engineering: Evaluating the stress on a material at a given load using the stress-strain curve.
• Economics: Evaluating the cost of a product at a given price using the demand curve.

Tips and Tricks

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When evaluating a function at a given value, it is essential to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate exponents next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

By following the order of operations, you can ensure that your calculations are accurate and reliable.

Common Mistakes

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When evaluating a function at a given value, some common mistakes to avoid include:

• Forgetting to substitute the value: Make sure to substitute the given value into the function.
• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) to ensure accurate calculations.
• Not simplifying the expression: Make sure to simplify the expression after substituting the value and following the order of operations.

By avoiding these common mistakes, you can ensure that your calculations are accurate and reliable.

Final Thoughts

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Evaluating a function at a given value is a fundamental concept in mathematics that has many practical applications in science and engineering. By following the order of operations and avoiding common mistakes, you can ensure that your calculations are accurate and reliable. In this article, we evaluated the function $h(x) = \frac{1}{3} \cdot 6^x$ at the given value $x = 2$ and obtained the result $h(2) = 12$.

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Introduction

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In our previous article, we evaluated the function $h(x) = \frac{1}{3} \cdot 6^x$ at the given value $x = 2$ and obtained the result $h(2) = 12$. In this article, we will answer some frequently asked questions about evaluating functions at given values.

Q&A

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Q: What is the purpose of evaluating a function at a given value?

A: The purpose of evaluating a function at a given value is to determine the output of the function for a specific input. This is useful in many areas of mathematics and science, such as physics, engineering, and economics.

Q: How do I evaluate a function at a given value?

A: To evaluate a function at a given value, you need to substitute the value into the function and simplify the expression. Make sure to follow the order of operations (PEMDAS) to ensure accurate calculations.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when evaluating an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate exponents next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

Q: What are some common mistakes to avoid when evaluating a function at a given value?

A: Some common mistakes to avoid when evaluating a function at a given value include:

• Forgetting to substitute the value: Make sure to substitute the given value into the function.
• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) to ensure accurate calculations.
• Not simplifying the expression: Make sure to simplify the expression after substituting the value and following the order of operations.

Q: How do I know if I have evaluated a function correctly?

A: To ensure that you have evaluated a function correctly, make sure to:

• Check your work: Double-check your calculations to ensure that you have followed the order of operations and simplified the expression correctly.
• Use a calculator: If you are unsure about your calculations, use a calculator to check your work.
• Ask for help: If you are still unsure, ask a teacher or tutor for help.

Q: What are some real-world applications of evaluating functions at given values?

A: Evaluating functions at given values has many practical applications in science and engineering, such as:

• Physics: Evaluating the position of an object at a given time using the equation of motion.
• Engineering: Evaluating the stress on a material at a given load using the stress-strain curve.
• Economics: Evaluating the cost of a product at a given price using the demand curve.

Conclusion

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Evaluating functions at given values is a fundamental concept in mathematics that has many practical applications in science and engineering. By following the order of operations and avoiding common mistakes, you can ensure that your calculations are accurate and reliable. In this article, we answered some frequently asked questions about evaluating functions at given values and provided some real-world applications of this concept.

Example Problems

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Problem 1: Evaluate the function $f(x) = 2x^2 + 3x - 1$ at $x = 4$.

Solution:

$f(4) = 2(4)^2 + 3(4) - 1$ $f(4) = 2(16) + 12 - 1$ $f(4) = 32 + 12 - 1$ $f(4) = 43$

Problem 2: Evaluate the function $g(x) = \frac{1}{x} + 2$ at $x = 3$.

Solution:

$g(3) = \frac{1}{3} + 2$ $g(3) = \frac{1}{3} + \frac{6}{3}$ $g(3) = \frac{7}{3}$

Final Thoughts

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Evaluating functions at given values is a fundamental concept in mathematics that has many practical applications in science and engineering. By following the order of operations and avoiding common mistakes, you can ensure that your calculations are accurate and reliable. In this article, we answered some frequently asked questions about evaluating functions at given values and provided some real-world applications of this concept.