Question

The function \(f(x)=\frac{2}{3} x+15\) is increasing on the interval \((-\infty, \infty)\)

Short answer

The function is increasing on $(-\bm{\tco - \bm{\tco}}.

Step-by-step solution

Identify the Type of Function

Recognize that the given function is a linear function of the form \(f(x) = ax + b\) where \(a = \frac{2}{3}\) and \(b = 15\).

Determine the Slope

The slope of the function is given by the coefficient of \(x\) in the linear function, which is \(a = \frac{2}{3}\).

Interpret the Slope

For a linear function, if the slope \(a\) is positive, the function is increasing over its entire domain. In this case, since \(\frac{2}{3}\) is positive, the function \(f(x)\) is increasing.

Conclusion

Since the slope \(\frac{2}{3}\) is positive, the function \(f(x) = \frac{2}{3} x + 15\) is increasing on the interval \((-\bm{\tcy} \bmin \bm{\bm{\tcy} \bmin \bm{\bm{\tcrow}}\).

Key concepts

Slope

The slope of a straight line is a measure of its steepness and direction. For the linear function given by the exercise, we have the formula \[f(x) = \frac{2}{3} x + 15theon.\]
The slope is represented by the coefficient of \(x\), which in this case is \(a = \frac{2}{3}\). The positive value of the slope \( \frac{2}{3} \) tells us that the function is increasing.

Slope can be thought of as the 'rise over run'. For every unit you move to the right along the x-axis, the function's value (y-value) increases by the slope amount. In this context, for every 3 units you move to the right on the x-axis, the function value rises 2 units.

Increasing Function

An increasing function is a function where, as the input (x) increases, the output (f(x)) also increases. In simpler terms, if the slope of the function is positive, then the function will always go upwards as you move from left to right on a graph.

Our function \(f(x) = \frac{2}{3} x + 15\) has a positive slope of \(\frac{2}{3}\), meaning that it will continually increase across its entire domain. This is why the function is stated to be 'increasing on the interval \((-fty, fty)\).

A practical way to visualize this is by imagining hiking up a hill; as you move forward, you keep climbing up.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) that will produce a valid output (f(x)). For a linear equation like \(f(x) = \frac{2}{3} x + 15\), the domain is all real numbers.

This function is defined for every real number because there are no restrictions on the values that x can take. Thus, the domain is expressed as \( (-fty, fty)\).

This means you can plug in any real number into the function for x and get a corresponding f(x) value.

Linear Equation

A linear equation is an equation that forms a straight line when graphed on a coordinate plane. The general form of a linear equation is \(y = mx + b\), where:

• \(m\) is the slope.
• \(b\) is the y-intercept (the point where the line crosses the y-axis).

In the exercise, the given linear equation is \(f(x) = \frac{2}{3} x + 15\). Here, the slope \(m\) is \(\frac{2}{3}\), and the y-intercept \(b\) is 15.

Understanding linear equations is fundamental since they are among the simplest and most commonly used functions. They are straightforward to work with and solve, making them an essential part of any basic algebra curriculum.