Problem 50 - 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. $$ \text { If } P(x)=5 x+1, \text { find } P(a), P(-x), \text { and } P(x+h) $$

Step-by-Step Solution

Answer

P(a) = 5a + 1; P(-x) = -5x + 1; P(x+h) = 5x + 5h + 1

Step 1: Understanding the function

The given polynomial function is \(P(x) = 5x + 1\). This is a linear polynomial function where each input \(x\) produces an output by multiplying \(x\) by 5 and then adding 1.

Step 2: Finding \(P(a)\)

To find \(P(a)\), substitute \(a\) for \(x\) in the function. Thus, \(P(a) = 5a + 1\).

Step 3: Finding \(P(-x)\)

To find \(P(-x)\), substitute \(-x\) for \(x\) in the function. So, \(P(-x) = 5(-x) + 1 = -5x + 1\).

Step 4: Finding \(P(x+h)\)

To find \(P(x+h)\), substitute \(x + h\) for \(x\) in the function. Thus, \(P(x+h) = 5(x+h) + 1 = 5x + 5h + 1\).

Key Concepts

Polynomial Evaluation

Polynomial evaluation is the process of determining the output of a polynomial function for a given input. In simpler terms, you are plugging values into the polynomial to find out what it equals. Polynomials are expressions made up of variables and coefficients, involving terms such as constants, single variables, and variables raised to powers.
For example, consider a polynomial function like \(P(x) = 5x + 1\). Evaluating this polynomial means testing how it behaves for specific inputs.

• Evaluate for \(x = a\): Replace \(x\) with \(a\) to find \(P(a) = 5a + 1\).
• Evaluate for \(x = -x\): Replace \(x\) with \(-x\) to find \(P(-x) = -5x + 1\).
• Evaluate for \(x = x+h\): Replace \(x\) with \(x+h\) to find \(P(x+h) = 5x + 5h + 1\).

Understanding polynomial evaluation helps in predicting the outputs for different inputs, which is a foundation for calculus and other mathematical fields.

Function Notation

Function notation is a way to express the relationship between the inputs of a function and its outputs. It provides a compact and precise format for writing functions that make it easier to handle complex calculations and communicate mathematical ideas.
The notation \(P(x)\) stands for a function \(P\) that depends on \(x\). The letter inside the parentheses is the variable or input. The expression on the other side of the equal sign shows what you do to that input. For \(P(x) = 5x + 1\), this means:

• The function name is \(P\), indicating it can be part of a series of functions like \(Q(x)\) or \(R(x)\).
• The input variable is \(x\), the placeholder that you can replace with a particular value or expression.
• The output rule is \(5x + 1\), showing how \(x\) is transformed into \(P(x)\).

Function notation is crucial because it allows us to systematically change inputs and see the effects on the outputs without rewriting the entire expression each time. It clarifies what we are calculating and how different inputs are treated.

Algebraic Substitution

Algebraic substitution is the technique of replacing a variable in an equation or function with a number, another variable, or an expression. This action allows you to evaluate functions and solve problems by simplifying expressions according to given criteria.
Let's break down this process with the example \(P(x) = 5x + 1\):

• Substitute \(a\) for \(x\): Replace \(x\) with \(a\) to transform the function into \(P(a) = 5a + 1\).
• Substitute \(-x\) for \(x\): Change \(x\) to \(-x\) and the expression becomes \(P(-x) = -5x + 1\).
• Substitute \(x+h\) for \(x\): By inserting \(x+h\) in place of \(x\), the function evaluates to \(P(x+h) = 5x + 5h + 1\).

Substitution is fundamental in algebra as it enables the manipulation and evaluation of expressions efficiently. It transforms functions into a more useful or simpler form without altering their inherent relationships.