Problem 52 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. $$g(a-9)$$

Step-by-Step Solution

Answer

\(g(a-9) = a^2 - 11a + 20\)

Step 1: 1. Write down the given function g(x)

First, let's write down the given function, \(g(x)=x^2+7x+2\).

Step 2: 2. Substitute (a-9) for x

Now, we will substitute the expression (a-9) in place of x in the function \(g(x)\). This will give us: \(g(a-9)=(a-9)^2+7(a-9)+2\)

Step 3: 3. Expand the expressions

We will now expand the expressions inside the parentheses: \((a-9)^2 = a^2 - 18a + 81\) \(7(a-9) = 7a - 63\)

Step 4: 4. Plug in the expanded expressions into g(a-9)

By substituting the expanded expressions into our equation for \(g(a-9)\), we get: \(g(a-9)=(a^2 - 18a + 81) + (7a - 63) + 2\)

Step 5: 5. Simplify the expression

Finally, we will combine like terms to simplify the expression: \(g(a-9) = a^2 - 18a + 81 + 7a - 63 + 2\) \(g(a-9) = a^2 - 11a + 20\) Thus, the simplified expression for \(g(a-9)\) is \(a^2 - 11a + 20\).

Key Concepts

Algebra

Algebra serves as the cornerstone of modern mathematics by providing a language to communicate mathematical ideas. It involves manipulating symbols and numbers to solve equations and understand patterns. In algebra, letters like \(x\) or \(a\) often represent unknown numbers in order to generalize formulas and procedures.

Basic operations in algebra include addition, subtraction, multiplication, and division, but they also extend into more complex manipulations such as factoring and expanding expressions. For example, in our exercise, we see operations like expanding \((a-9)^2\) and combining like terms. These techniques allow us to transform functions and expressions into simpler or more useful forms.

Understanding algebra is crucial for many areas of math and science as it develops problem-solving skills and logical thinking. It also forms the basis for more advanced topics like calculus and linear algebra.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of the variable \(x\), each multiplied by a coefficient. They can range from simple linear functions to more complex forms like quadratics or cubics. The polynomial functions take the form \(a_n x^n + a_{n-1} x^{n-1} + \, \ldots \, + a_1 x + a_0\).

In our exercise, \(g(x)=x^2+7x+2\) is a quadratic polynomial function, which is a second-degree polynomial. This means that the highest power of \(x\) is 2. Polynomial functions are continuous and smooth graphs, which makes them very useful in modeling real-life situations.

Understanding polynomial functions is essential because they provide a powerful framework for approximation in mathematical modeling. They allow mathematicians and scientists to predict and interpret various phenomena by fitting them into polynomial models.

Function Evaluation

Function evaluation is the process of finding the output of a function given an input value. In more formal terms, it involves substituting specific numbers or expressions into a function. This process is critical for determining how different inputs affect the outcomes in mathematical functions.

Our step-by-step solution involved evaluating \(g(a-9)\), which means substituting \(a-9\) for \(x\) in the polynomial function \(g(x)\). Here are key components of function evaluation:

• Identify the function and its variable. For instance, \(g(x)\) is the function where \(x\) is the variable.
• Substitute the specified value or expression into the function. Replace \(x\) with \(a-9\) to find \(g(a-9)\).
• Simplify the expression using algebraic techniques, such as expanding and combining like terms, to arrive at the final result.

Function evaluation helps verify the behavior of functions at specific points and is essential in many mathematical applications, including calculus and engineering. It provides insights into how a function behaves and responds to different inputs.