Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function at each specified value of the independent variable and simplify. $$g(y)=7-3 y$$ (a) \(g(0)\) (b) \(g\left(\frac{7}{3}\right)\) (c) \(g(s+5)\)

Step-by-Step Solution

Answer

(a) g(0) = 7\n(b) g(\frac{7}{3}) = 0\n(c) g(s+5) = -3s - 8

Step 1: Evaluate \(g(0)\)

Substitute \(y = 0\) into the function \(g(y)=7-3 y\):\n\n\(g(0) = 7 - (3 * 0)\)\n\nThe result is 7 because \(3 * 0\) equals 0.

Step 2: Evaluate \(g(\frac{7}{3})\)

Substitute \(y = \frac{7}{3}\) into the function \(g(y)=7-3 y\):\n\n\(g(\frac{7}{3}) = 7 - 3 * (\frac{7}{3})\)\n\nThe result is 0 because \(3 * \frac{7}{3}\) equals 7.

Step 3: Evaluate \(g(s+5)\)

Substitute \(y = s + 5\) into the function \(g(y)=7-3 y\):\n\n\(g(s+5) = 7 - 3 * (s + 5)\)\n\nSimplify the equation to get \(7 - 3s - 15\). \n\nFurther simplify to get \(g(s+5) = -3s - 8\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. They can represent relationships or equations in mathematics. In the problem we are exploring, the function is given as an algebraic expression:

• **Variables**: A symbol, like "y", that represents an unknown value or can take different values.
• **Operations**: These include addition, subtraction, multiplication, and division, which connect the numbers and variables.

In the expression \( g(y) = 7 - 3y \), the variable y is multiplied by 3 and then subtracted from 7. Understanding algebraic expressions is key to solving various problems in math, as they allow you to model real-life situations and perform calculations on them by altering the variable values.

Substitution Method

The substitution method is a straightforward way to evaluate expressions, particularly functions. It involves replacing the variables in the expression with given specific values. Here's how we used substitution in the original exercise:

• **Identify**: First, identify which variable needs replacement. In our case, this variable was "y."
• **Insert**: Directly substitute the identified value into the expression, where the variable appears.
• **Calculate**: Perform any operations necessary after substitution to find the result.
• **Simplify**: Finally, simplify the result by performing all arithmetic operations correctly.

This method is particularly useful with functions, as functions, by definition, relate inputs with outputs. By substituting different values, you can explore the behavior of the function across different inputs.

Simplification

Simplification in algebra is about making expressions as easy to understand and work with as possible. Once you have substituted values into a function, you'll often need to simplify the result. Let's break down the steps:

• **Combine Like Terms**: Terms that have the same variable part are combined. For instance, combining \(-3s\) and constants.
• **Perform Arithmetic Operations**: Carry out all additions, subtractions, multiplications, and divisions indicated in the expression.
• **Reduce Fractions**: If there are any fractions, reduce them to their simplest form.

In the example provided, after replacing the variable in the function with a given value, we reduced the expressions to forms like \(g(s + 5) = -3s - 8\). This ensures the result is clean and ready for interpretation or further computation.

Linear Functions

Linear functions are one of the simplest types of functions you will encounter in algebra. They form straight lines when graphed, and they have the form \( f(x) = mx + b \):

• **Constant Change**: The variable x is increased or decreased by a constant factor, given by the slope "m".
• **Y-Intercept**: The "b" in the equation indicates where the line crosses the y-axis.

In our exercise, the function \( g(y) = 7 - 3y \) is a linear function:

• **Slope**: The coefficient of y, which is -3, indicates the line's steepness and direction.
• **Intercept**: The constant, 7, gives the point where the line crosses the y-axis.

Understanding the form and behavior of linear functions is crucial, as they model many real-world phenomena where change is constant. They are foundational for understanding more complex relationships in mathematics.