Problem 24 - Algebra 2 and Trigonometry

Question

If \(\mathrm{p}(x)=2\) and \(\mathrm{q}(x)=x+2,\) find \(\mathrm{p}(\mathrm{q}(5))\) and \(\mathrm{q}(\mathrm{p}(5))\)

Step-by-Step Solution

Answer

\( \mathrm{p}(\mathrm{q}(5)) = 2 \) and \( \mathrm{q}(\mathrm{p}(5)) = 4 \).

Step 1: Identify Functions

We are given two functions: - \( \mathrm{p}(x) = 2 \) which is a constant function.- \( \mathrm{q}(x) = x+2 \) which is a linear function.Our task is to find \( \mathrm{p} \left( \mathrm{q}(5) \right) \) and \( \mathrm{q} \left( \mathrm{p}(5) \right) \).

Step 2: Evaluate \( \mathrm{q}(5) \)

Substitute \( x = 5 \) into the function \( \mathrm{q}(x) = x + 2 \): \[ \mathrm{q}(5) = 5 + 2 = 7 \].

Step 3: Evaluate \( \mathrm{p} \left( 7 \right) \)

Since the function \( \mathrm{p}(x) = 2 \) is a constant function, it is independent of the input. So, \[ \mathrm{p} \left( 7 \right) = 2 \].

Step 4: Evaluate \( \mathrm{p}(5) \)

Substitute \( x = 5 \) into the function \( \mathrm{p}(x) = 2 \): \[ \mathrm{p}(5) = 2 \].

Step 5: Evaluate \( \mathrm{q}(2) \)

Substitute \( x = 2 \) into the function \( \mathrm{q}(x) = x + 2 \): \[ \mathrm{q}(2) = 2 + 2 = 4 \].

Key Concepts

Constant Function

A constant function is one where the output value always remains the same, regardless of the input value. This can be represented as \( f(x) = c \), where \( c \) is a constant number.
Here, the function \( p(x) = 2 \) is a classic example of a constant function. No matter what value of \( x \) you use as input for \( p(x) \), the result will always be 2.
Some key characteristics of constant functions include:

• They are graphed as straight horizontal lines in the Cartesian coordinate system.
• The slope of a constant function's graph is 0.
• They are the simplest kind of function to evaluate due to their non-dependence on \( x \).

Constant functions are helpful in simplifying expressions and calculations because they provide a consistent output. For example, in our exercise, no matter the input for \( p(x) \), we already know the output, simplifying the process of evaluating nested functions.

Linear Function

Linear functions are represented by the equation \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
In this exercise, \( q(x) = x + 2 \) is an example of a linear function, where the slope \( m \) is 1 (you go up one step for every one step you go to the right) and the y-intercept \( b \) is 2.
Some essential features of linear functions are:

• The graph is a straight line.
• The slope \( m \) indicates how steep the line is and the direction it goes (positive or negative).
• Linear functions grow at a constant rate.

Linear functions are widely used for modeling relationships where there is a consistent rate of change. In this exercise, we used \( q(x) \) to evaluate different inputs and observe how it transforms them by consistently adding 2.

Evaluating Functions

Evaluating functions means finding the output of a function for a specific input, essentially "plugging in" a value and calculating the result. It’s a critical skill in handling more complex mathematical concepts like function composition.
Let's break down the evaluation process:

• Identify the function and its parameters.
• Substitute the given input value into the function.
• Perform any necessary operations to solve for the output.

In our original problem, evaluating \( q(5) \) meant substituting 5 for \( x \) in the function \( q(x) = x + 2 \), yielding \( q(5) = 7 \). Similarly, for constant function \( p(x) = 2 \), regardless of the input \( x \), the output is always 2.
Function composition, which combines two or more functions, relies heavily on accurate evaluation. By first solving \( q(5) \) then \( p(7) \), we explore how functions interact, helping us understand the "composition" or combined effect of two functions: \( p(q(5)) \). Understanding these evaluation steps enhances problem-solving and mathematical reasoning significantly.