Evaluating Functions And Solving Equations Consider The Function G(x) = 5x + 14
Function Definition
We are given the function g(x) = 5x + 14. This is a linear function, where for any input value x, the function g multiplies it by 5 and then adds 14 to the result. This type of function is fundamental in mathematics and appears in various applications, from simple algebraic equations to more complex calculus problems. Understanding how to evaluate such functions for different values of x and solving for x when g(x) is known are crucial skills in algebra and beyond. In this comprehensive guide, we will explore how to evaluate this function for given values of x and how to find the value of x that corresponds to a specific output of the function. This includes substituting the given values into the function and performing the arithmetic operations to find the output, as well as setting the function equal to a given value and solving the resulting equation to find the input. This process not only reinforces basic algebraic skills but also lays the foundation for understanding more complex mathematical concepts. By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical problems that involve functions and their evaluations. The ability to work with functions like g(x) = 5x + 14 is a stepping stone to understanding more advanced topics such as calculus, where functions and their behavior are central to the study of rates of change and accumulation. The function g(x) = 5x + 14 is a linear function because it can be written in the form g(x) = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is 5, and the y-intercept is 14. The slope indicates how much the function's output changes for each unit increase in the input, while the y-intercept is the value of the function when x = 0. Linear functions are characterized by their constant rate of change, which means that the output changes by the same amount for each unit change in the input. This property makes them predictable and easy to analyze, which is why they are often used as a starting point for modeling real-world phenomena. For example, a linear function could be used to model the cost of a service that charges a fixed fee plus an hourly rate, or the distance traveled by a car moving at a constant speed. Understanding linear functions is essential for building a solid foundation in algebra and for applying mathematical concepts to solve practical problems. The ability to work with linear functions is also crucial for understanding more complex types of functions, such as quadratic, exponential, and trigonometric functions. These functions exhibit different behaviors and have their own unique properties, but the basic principles of function evaluation and equation solving remain the same. Therefore, mastering linear functions is a key step in becoming proficient in mathematics. This knowledge will enable you to tackle more advanced topics and to apply mathematical concepts to a wide range of real-world situations.
(a) Find g(2)
To find g(2), we substitute x = 2 into the function g(x) = 5x + 14. This means we replace every instance of x in the function's formula with the value 2. This substitution is a fundamental step in evaluating functions, allowing us to determine the output of the function for a specific input value. The process of substitution is not limited to numerical values; we can also substitute algebraic expressions for x, which is a common technique in simplifying or manipulating functions. However, in this case, we are dealing with a simple numerical substitution, which allows us to directly calculate the value of the function. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial in ensuring that we arrive at the correct answer. In this case, we first perform the multiplication (5 * 2) and then the addition (10 + 14). Understanding the order of operations is essential for accurate mathematical calculations, especially when dealing with more complex expressions involving multiple operations. Mistakes in the order of operations can lead to incorrect results, so it's important to follow the established rules. Once we have substituted x = 2 into the function, we perform the arithmetic operations to simplify the expression. This involves multiplying 5 by 2, which gives us 10, and then adding 14 to the result. The arithmetic operations are the building blocks of mathematical calculations, and proficiency in these operations is essential for success in mathematics. Addition, subtraction, multiplication, and division are the fundamental operations, and a strong grasp of these operations is necessary for tackling more complex mathematical problems. The ability to perform arithmetic operations accurately and efficiently is a valuable skill that will benefit you in various aspects of life, not just in mathematics. After performing the arithmetic operations, we arrive at the final value of g(2). This value represents the output of the function when the input is 2. The output of a function is the result obtained after applying the function's rule to the input value. In this case, the function's rule is to multiply the input by 5 and then add 14. The output value is crucial in understanding the behavior of the function and its relationship to the input value. By evaluating the function for different input values, we can gain insights into the function's properties, such as its range, its intercepts, and its increasing or decreasing behavior. The ability to interpret the output of a function in the context of a problem is an important skill in mathematical modeling and problem-solving. In summary, finding g(2) involves substituting x = 2 into the function g(x) = 5x + 14, performing the arithmetic operations, and obtaining the final value. This process demonstrates the fundamental principles of function evaluation and highlights the importance of substitution and the order of operations. The result, g(2) = 24, is a specific point on the graph of the function, representing the y-coordinate when the x-coordinate is 2. This point provides valuable information about the function's behavior and its position in the coordinate plane. The ability to evaluate functions for specific input values is a crucial skill in mathematics, as it allows us to analyze and understand the behavior of functions and their applications in various fields.
The given solution is:
g(2) = 24
(b) Find g(-11)
Now, let's find g(-11). This is similar to the previous part, but this time, we are substituting a negative value, x = -11, into the function g(x) = 5x + 14. Working with negative numbers is a fundamental skill in algebra and arithmetic. It's essential to understand how negative numbers interact with arithmetic operations, such as multiplication and addition. Substituting a negative value into a function can sometimes lead to unexpected results, so it's crucial to pay close attention to the signs and follow the correct order of operations. The process of substituting a negative value is identical to substituting a positive value; we simply replace every instance of x in the function's formula with the value -11. However, we need to be extra careful when dealing with negative numbers, as errors in sign can lead to incorrect results. The rules for multiplying and adding negative numbers are essential to remember: a negative number multiplied by a positive number is negative, and a negative number plus a positive number can be positive, negative, or zero, depending on the magnitudes of the numbers. After substituting x = -11 into the function, we again follow the order of operations (PEMDAS) to simplify the expression. This means we first perform the multiplication (5 * -11) and then the addition (-55 + 14). Paying close attention to the signs is crucial in this step, as an error in the sign can lead to an incorrect result. The multiplication of 5 and -11 results in -55, which is a negative number. This is because a positive number multiplied by a negative number is always negative. Then, we add 14 to -55. This is where understanding the rules for adding numbers with different signs is important. When adding a negative number and a positive number, we essentially subtract the smaller magnitude from the larger magnitude and keep the sign of the number with the larger magnitude. In this case, the magnitude of -55 is greater than the magnitude of 14, so the result will be negative. Performing the addition -55 + 14 involves subtracting 14 from 55, which gives us 41. Since the larger magnitude was negative, the final result is -41. This is the value of g(-11). The negative result indicates that when the input to the function is -11, the output is -41. This point, (-11, -41), lies in the third quadrant of the coordinate plane, where both x and y coordinates are negative. Understanding the sign of the output is crucial in interpreting the behavior of the function. A negative output indicates that the function's value is below the x-axis, while a positive output indicates that the function's value is above the x-axis. In summary, finding g(-11) involves substituting x = -11 into the function g(x) = 5x + 14, carefully performing the arithmetic operations with negative numbers, and obtaining the final value. This process reinforces the importance of understanding negative numbers and the rules for arithmetic operations. The result, g(-11) = -41, is a specific point on the graph of the function, representing the y-coordinate when the x-coordinate is -11. This point provides valuable information about the function's behavior and its position in the coordinate plane. The ability to evaluate functions for negative input values is a crucial skill in mathematics, as it allows us to analyze and understand the behavior of functions over a wider range of inputs.
The given solution is:
g(-11) = -41
(c) Find x such that g(x) = 49
This part is different. We are asked to find the value of x such that g(x) = 49. This is the reverse of the previous two parts, where we were given the value of x and asked to find g(x). Now, we are given the value of g(x) and asked to find x. This involves solving an equation, which is a fundamental skill in algebra. Solving equations is a cornerstone of algebra and is essential for solving a wide range of mathematical problems. An equation is a statement that two expressions are equal, and solving an equation means finding the value(s) of the variable(s) that make the equation true. In this case, we have the equation 5x + 14 = 49, and we need to find the value of x that satisfies this equation. The process of solving an equation involves isolating the variable on one side of the equation by performing the same operations on both sides. This ensures that the equation remains balanced and that the solution we obtain is valid. The operations we can perform include addition, subtraction, multiplication, and division, as well as more complex operations such as taking the square root or raising to a power. To find x, we set the function g(x) = 5x + 14 equal to 49: 5x + 14 = 49. This creates an equation that we can solve for x. The equation 5x + 14 = 49 represents a linear equation in one variable, x. Linear equations are the simplest type of equations to solve, and they have a unique solution, provided the coefficient of the variable is not zero. The general form of a linear equation is ax + b = c, where a, b, and c are constants and x is the variable. To solve a linear equation, we typically use a series of algebraic manipulations to isolate the variable on one side of the equation. The goal is to get x by itself on one side of the equation, with a constant on the other side. First, we subtract 14 from both sides of the equation: 5x + 14 - 14 = 49 - 14. This is a valid algebraic manipulation because subtracting the same value from both sides of an equation does not change its solution. The subtraction eliminates the constant term on the left side of the equation, bringing us closer to isolating x. Subtracting 14 from both sides simplifies the equation to 5x = 35. Now, we have a simpler equation where the variable x is multiplied by a constant. To isolate x, we need to perform the inverse operation of multiplication, which is division. Next, we divide both sides of the equation by 5: 5x / 5 = 35 / 5. This is another valid algebraic manipulation because dividing both sides of an equation by the same non-zero value does not change its solution. The division isolates x on the left side of the equation. Dividing both sides by 5 gives us x = 7. This is the solution to the equation 5x + 14 = 49. This means that when x is equal to 7, the value of the function g(x) is 49. We can verify this by substituting x = 7 into the function g(x) = 5x + 14: g(7) = 5(7) + 14 = 35 + 14 = 49. This confirms that our solution is correct. The solution x = 7 represents a specific point on the graph of the function, where the y-coordinate is 49. This point provides valuable information about the function's behavior and its relationship to the x-axis. In summary, finding x such that g(x) = 49 involves setting the function equal to 49, solving the resulting equation using algebraic manipulations, and obtaining the value of x. This process demonstrates the fundamental principles of equation solving and highlights the importance of inverse operations in isolating the variable. The solution, x = 7, is the input value that produces an output of 49 from the function g(x) = 5x + 14. This skill is crucial in many mathematical and real-world applications, where we need to determine the input that produces a desired output. This skill is a crucial part of algebraic problem-solving, showing how to work backwards from an output to find the corresponding input.
Therefore,
x = 7
