Question

The two functions share either an inside function or an outside function. Which is it? Describe the shared function. $$ y=(2 x+1)^{3} \text { and } y=\frac{1}{\sqrt{2 x+1}} $$

Short answer

Answer: The shared function between the given functions is \((2x + 1)\), which is a linear function with a slope of 2 and a y-intercept of 1.

Step-by-step solution

Identify the structure of given functions

Write down the given functions in separate equations. $$ y_1 = (2x + 1)^3 $$ and $$ y_2 = \frac{1}{\sqrt{2x + 1}} $$

Analyze the functions

Now, let's analyze the structure of each function. The first function is a cubic function. The second function is a fractional function with a square root in the denominator. Notice that both functions have \((2x + 1)\) as a part of their structure. In \(y_1\), it is raised to the power of 3, and in \(y_2\), it is inside the square root term in the denominator.

Determine the shared function

As we observed in the previous step, both functions contain \((2x+1)\). This is the shared function. The shared function is an inside function in both cases.

Describe the shared function

The shared function is: $$ (2x + 1) $$ This function is a linear function with a slope of 2 and a y-intercept of 1. It is inside another function in both given equations. In the first function, it is inside a cubic function, and in the second function, it is inside a square root in the denominator.

Key concepts

Cubic Function

A cubic function is a type of polynomial function that has the highest power of three. You can usually identify it by looking at the exponent on the variable, like in the function \( y_1 = (2x + 1)^3 \). The structure here indicates a cubic relationship. Cubic functions have several interesting features:

• They can have up to three real roots, or "x-intercepts."
• They reach a special kind of curve called an "inflection point," where the concavity changes.
• The overall shape can look like an elongated S or reverse S, depending on the coefficients and terms involved.

In the context of the given exercise, the cubic function has \((2x + 1)\) as the base which is raised to the power of 3. This makes \((2x + 1)\) an inside function and tells us we're dealing with the more substantial inflection aspect of the cubic function. Understanding cubic functions is crucial because they exhibit the nature of diminishing returns or accelerations and decelerations, something often found in physics and economics.

Square Root Function

A square root function is a type of radical function that generally involves finding the square root of a variable. Its typical form can be seen as \( y = \frac{1}{\sqrt{x}} \), which matches \( y_2 = \frac{1}{\sqrt{2x + 1}} \) in our context.Key aspects about square root functions:

• They are only defined for non-negative numbers unless extended to complex numbers.
• Their graphs are always increasing or remaining constant as they have a restricted domain.
• The output values grow smaller as the input value becomes much larger.

In the exercise, \( y_2 \) contains a square root function in the denominator. This creates a scenario where the shared inside function, \((2x + 1)\), dictates how the inputs are modified before undergoing further transformation via the square root. Therefore, much of the behavior and output of the square root function come down to how the inside linear function modifies the input variables. Square roots are useful for modeling situations where growth slows over time, like in sound intensity or luminosity.

Linear Function

A linear function is the simplest type of polynomial, represented by functions of the form \( y = mx + b \). In our case, the linear function is \( y = 2x + 1 \), which is shared by both the cubic and square root functions from the exercise.Here’s what makes linear functions stand out:

• They have a constant rate of change or slope.
• Graphs of these functions always produce a straight line.
• The slope \( m \) in \( y = mx + b \) tells you how steep the line is; a positive slope means an upward trend, and a negative slope indicates a downward trend.

In the step-by-step solution, \( (2x + 1) \) was recognized as the shared function within both given functions. With a slope of 2, the shared function forms the core manipulation for both the cubic and square root transformations. This makes it crucial because changes to this linear part have direct and predictable effects on the outputs of both composite functions. Linear functions are straightforward and widely used to approximate conditions with consistent direct proportionality and relationships.