Problem 9 - Algebra A Combined Function

Question

If \(P(x)=x^{3}+2 x-3\) and \(Q(x)=7 x+5,\) find each function value. $$ Q(-3) $$

Step-by-Step Solution

Answer

\(Q(-3) = -16\).

Step 1: Identify the function for the given input

We are asked to find the function value for \(Q(-3)\). The function \(Q(x)\) is given by \(Q(x) = 7x + 5\). Our task is to find \(Q(-3)\) by substituting \(-3\) for \(x\) in \(Q(x)\).

Step 2: Substitute the input into the function

Substitute \(-3\) into \(Q(x)\). This gives us:\[ Q(-3) = 7(-3) + 5 \]

Step 3: Simplify the expression

Simplify the expression by performing the multiplication and addition: \[ Q(-3) = 7(-3) + 5 = -21 + 5 \].

Step 4: Calculate the final result

Compute the final result by adding the numbers together: \(-21 + 5 = -16\). Thus, \(Q(-3) = -16\).

Key Concepts

Polynomial Functions

Polynomial functions are a type of algebraic expression that consist of variables and coefficients. They include terms with variables raised to whole number powers. Typically, these terms are added or subtracted together. Simple polynomial functions might look like this: \( P(x) = x^3 + 2x - 3 \). Here are some important points about polynomial functions:

• The highest power of the variable (exponential term) determines the degree of the polynomial. In the example above, \( x^3 \) is the highest power, so it is a cubic polynomial.
• The coefficient is the number that multiplies the variable, such as the \( 2 \) in \( 2x \) or the hidden \( 1 \) before \( x^3 \).
• Constant terms are standalone numbers in the polynomial, like \( -3 \) in the example.

Understanding polynomial functions is fundamental in algebra, as they are the basis for a wide range of mathematical problems and solutions.

Function Evaluation

Function evaluation is the process of determining the output of a function given a specific input. Each function operates like a machine, taking an input and returning an output based on its formula. For example, in the exercise where \( Q(x) = 7x + 5 \), you're tasked with evaluating \( Q(-3) \).

• First, you need to identify the input values. In this case, it’s \(-3\).
• Second, substitute this input into the function, replacing all occurrences of \( x \) with \(-3\).
• Math can then be done to find the output, which is \( Q(-3) = 7(-3) + 5 \).
• The final result after calculation is \( -16 \).

Function evaluation is crucial for solving equations and understanding the behavior of functions across different points. It helps in graph plotting and real-world problem modeling.

Substitution Method

The substitution method is a technique used to replace variables in expressions with given values. This method lies at the heart of evaluating functions and solving equations. It involves the following steps:

• Identify the variable to be substituted. For function evaluations, it's usually the input value that replaces the function’s main variable, like \( x \) in \( Q(x) \).
• Replace every instance of the identified variable in the expression with the given value. In our example, replace \( x \) with \(-3\) to find \( Q(-3) \).
• Simplify the expression by performing basic arithmetic operations. In the case of \( Q(x) = 7x + 5 \) and substituting \(-3\), first calculate \( 7(-3) \) and add \( 5 \).
• Finally, solve to find the expression's value, which gives the answer, such as \( -16 \) in this exercise.

The substitution method is a staple of algebra that helps simplify complex expressions and find solutions for specific variable values. It aids in learning how inputs affect outputs in mathematical functions and equations.