Problem 35 - Beginning and Intermediate Algebra

Question

Let \(f(x)=3 x-7\) and \(g(x)=x^{2}-4 x-9 .\) Find each of the following and simplify. $$f(c+4)$$

Step-by-Step Solution

Answer

\(f(c+4) = 3c + 5\)

Step 1: Write down the given functions

The given functions are: \(f(x) = 3x - 7\)

Step 2: Substitute c+4 into the function f(x)

To find f(c+4) we substitute x with (c+4) in f(x) function: \(f(c+4) = 3(c+4) - 7\)

Step 3: Simplify the expression

Now, distribute the 3 to both terms inside the parenthesis and then subtract 7: \(f(c+4) = 3c + 12 - 7\) Combine the constants 12 and -7: \(f(c+4) = 3c + 5\) So, the simplified expression of \(f(c+4)\) is \(3c + 5\).

Key Concepts

Function Substitution

Function substitution is an essential concept in algebra that involves replacing variables in a function with either numbers or other expressions. This is a powerful tool as it allows us to evaluate functions and transform them into simpler forms.

To use function substitution effectively:

• Identify the function given to you. For example, if you have a function like \(f(x) = 3x - 7\), understand the role of the variable \(x\).
• Substitute the expression or value in place of the variable. If asked to evaluate \(f(c+4)\), replace every occurrence of \(x\) in \(f(x)\) with \(c+4\).
• Once substituted, the function will transform and become a new expression depending on the substituted value.

Understanding substitution is the key step for both evaluating functions and transforming them for further simplification.

Polynomial Functions

Polynomial functions, like \(g(x)=x^{2}-4x-9\), play a significant role in algebra and are at the heart of many calculations. A polynomial function is composed of terms called monomials, which are sums of variables raised to whole number powers, multiplied by coefficients.

Main components of polynomial functions include:

• Terms: Each polynomial is made up of terms. In \(g(x) = x^2 - 4x - 9\), the terms are \(x^2\), \(-4x\), and \(-9\).
• Degree: The highest power of the variable in the polynomial. Here, the largest exponent is 2, so \(g(x)\) is a quadratic function (second-degree).
• Coefficients: These are the numbers in front of the variables. In our example, the coefficient of \(x^2\) is 1, for \(-4x\) it is -4, and the last term, \(-9\), is a constant.

By mastering polynomial functions, you can tackle a broad range of algebraic and real-world problems.

Simplification of Expressions

Simplification of expressions is a fundamental process in algebra that aims to reduce expressions to their simplest forms. This involves combining like terms and using arithmetic operations to make expressions more manageable and easier to interpret.

Here's how to simplify an expression like \(f(c+4) = 3(c+4) - 7\):

• Distribute: Apply the distributive property to eliminate parentheses: \(3(c+4) - 7 = 3c + 12 - 7\).
• Combine Like Terms: Add or subtract terms that have the same variables or are constants. The terms 12 and -7 are constants and can be combined to give \(3c + 5\).
• Through simplification, not only does the expression become easier to work with, but it also reveals insights into the function's behavior.

Mastering this skill will enhance your ability to solve algebraic expressions efficiently and accurately.