Problem 49 - Algebra A Combined Function

Question

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7,\) find each function. $$ \text { If } P(x)=2 x-3, \text { find } P(a), P(-x), \text { and } P(x+h) $$

Step-by-Step Solution

Answer

P(a) = 2a - 3, P(-x) = -2x - 3, P(x+h) = 2x + 2h - 3.

Step 1: Understanding the Function

We start by understanding the given function. We have the function \( P(x) = 2x - 3 \). This function is defined for any value of \( x \) and represents a linear polynomial.

Step 2: Evaluating P(a)

To find \( P(a) \), substitute \( a \) into the function \( P(x) \). This gives us: \[ P(a) = 2a - 3 \] Thus, \( P(a) \) is the value of the function when \( x = a \).

Step 3: Evaluating P(-x)

To find \( P(-x) \), substitute \( -x \) into the function \( P(x) \). This gives us: \[ P(-x) = 2(-x) - 3 = -2x - 3 \] So, \( P(-x) = -2x - 3 \).

Step 4: Evaluating P(x+h)

For \( P(x+h) \), substitute \( x + h \) into the function \( P(x) \). This gives: \[ P(x+h) = 2(x+h) - 3 = 2x + 2h - 3 \] Therefore, \( P(x+h) = 2x + 2h - 3 \).

Key Concepts

polynomials

Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. A polynomial can be as simple as a single variable term like \(x\), or a combination like \(2x^2 + 3x - 5\). The key characteristic of polynomials is that these expressions contain variables raised to whole number exponents.
Some polynomials include:

• Monomial: A polynomial with just one term. Example: \(4x^3\).
• Binomial: A polynomial with two terms. Example: \(3x^2 - 2\).
• Trinomial: A polynomial with three terms. Example: \(x^2 + 5x + 6\).

Polynomials are used to represent and solve a variety of problems in math and science. They can describe routes, areas, volumes, and more. Recognizing how terms combine helps in simplifying, expanding, and factoring expressions to solve equations. Understanding polynomials deeply aids in grasping more complex algebraic concepts.

linear function

A linear function is a type of polynomial that is the simplest among them. It typically has the form \(f(x) = mx + b\), where:

• \(m\) is the slope, showing how steep the line is.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

For example, the function \(P(x) = 2x - 3\) is linear because it fits this form. The slope is 2, and the y-intercept is -3. Linear functions graph to a straight line, hence the name "linear."
They are used in a variety of fields to show relationships with a constant rate of change. This means that for each unit increase in \(x\), the output \(y\) increases by \(m\) units. Identifying the slope and y-intercept helps to predict and analyze patterns or trends in data.

function evaluation

Function evaluation is a fundamental concept in algebra that involves calculating the output of a function for a specific input. It means "plugging in" given values for \(x\) in the function to get the corresponding \(y\) or output value.
For example, with the linear function \(P(x)=2x-3\), to evaluate \(P(a)\), replace \(x\) with \(a\):

• \(P(a) = 2a - 3\)

Similarly, for \(P(-x)\), replace \(x\) with \(-x\):

• \(P(-x) = -2x - 3\)

For \(P(x+h)\), substitute \(x+h\) into the function:

• \(P(x+h) = 2x + 2h - 3\)

Understanding how function evaluation works is crucial for solving equations, analyzing behavior of functions, and working through calculus concepts. It helps in predicting output and associating real-world phenomena with mathematical models.