Question

Write \(y\) as a function of \(x,\) where the value of \(y\) is equal to the given expression.Two-thirds of the amount by which \(x\) exceeds 4.

Short answer

The function of x is written as \( y = \frac{2}{3}(x - 4) \).

Step-by-step solution

Identify the expression

Understand that the task involves creating an expression for 'two-thirds of the amount by which x exceeds 4'. This implies a two-step operation where you first find the amount by which x is greater than 4, and then take two-thirds of this amount.

Calculate the amount by which x exceeds 4

To find out by how much x exceeds 4, subtract 4 from x. The expression for this is: x - 4.

Take two-thirds of the excess amount

Now, take two-thirds of the amount calculated in the previous step. Multiply the result of x - 4 by 2/3 to get this: \( y = \frac{2}{3}(x - 4) \).

Key concepts

Algebraic Expressions

Understanding algebraic expressions is crucial for solving many types of mathematical problems. An algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. For instance, in the exercise given, the expression 'two-thirds of the amount by which x exceeds 4' can be translated into an algebraic expression.

To break this down, we first identify the key parts: 'amount by which x exceeds 4' and 'two-thirds of'. To find how much x exceeds 4, we simply compute the difference, represented as \( x - 4 \). The next part requires us to take two-thirds of that difference, which we express as a multiplication by the fraction \(\frac{2}{3}\). Thus, we combine these two steps to create the algebraic expression \(\frac{2}{3}(x - 4)\). This expression encapsulates the entire operation in mathematical terms and is an essential skill in algebra.

Function Notation

Function notation is a streamlined way of writing an equation that defines a relationship between two variables. It is commonly represented as 'f(x)' which reads as 'f of x' where 'f' denotes the function, and 'x' is the input to the function. In the context of the exercise, we are asked to write 'y' as a function of 'x'.

To express the given algebraic expression as a function, we use function notation, resulting in \( y = f(x) = \frac{2}{3}(x - 4) \). Here, the function '(f)' is not explicitly mentioned, but it's understood that 'y' is the output when 'x' is the input. Using function notation makes it easier to identify and work with functions, especially when dealing with multiple functions or complex equations.

Linear Functions

A linear function is a type of function that creates a straight line when graphed on a coordinate plane. It is characterized by a constant rate of change, which can be thought of as the slope of the line. The general form of a linear function is \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept.

Looking at our function \( y = \frac{2}{3}(x - 4) \), we can see that it is indeed a linear function. Here, \( \frac{2}{3} \) is the slope which indicates that for every one unit increase in 'x', 'y' increases by two-thirds of a unit. The term '-4' is part of the y-intercept; however, it's multiplied by the slope in this function. To find the y-intercept, we would set 'x' to zero and compute the value of 'y'. In this case, the y-intercept is \( \frac{2}{3} \times -4 \) which simplifies to \( -\frac{8}{3} \). Linear functions such as this are foundational in algebra and appear in various real-world applications, from calculating speed to predicting profits.