Problem 28 - 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. $$f(0)$$

Step-by-Step Solution

Answer

The short answer for the problem is: \(f(0) = -7\).

Step 1: Identify f(x)

We are given the function $$f(x) = 3x - 7$$ and we are asked to find $$f(0)$$.

Step 2: Substitute x with 0 in f(x)

Replace x in f(x) with 0: $$f(0) = 3(0) - 7$$

Step 3: Simplify the expression

Now, simplify the expression by performing the multiplication and subtraction: $$f(0) = 0 - 7$$

Step 4: Write the final answer

The value of the function at x = 0 is: $$f(0) = -7$$

Key Concepts

Linear Functions

Linear functions like \( f(x) = 3x - 7 \) are one of the most fundamental concepts in mathematics. A linear function is a type of function that creates a straight line when plotted on a graph. It is defined by an equation of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

• In the equation \( f(x) = 3x - 7 \), \( 3 \) is called the slope, which determines the steepness of the line.
• The constant \( -7 \) is the y-intercept, which is the point where the line crosses the y-axis.

Linear functions are easy to evaluate at any given point \( x \) by simply substituting \( x \) into the equation and solving.

Substitution Method

The substitution method is a straightforward technique used to find the value of a function for a specific input. You replace the variable with the given number and perform the necessary arithmetic.

• For example, to find \( f(0) \) for \( f(x) = 3x - 7 \), you substitute 0 for \( x \).
• This gives you the expression \( 3(0) - 7 \).
• Once substituted, you just need to simplify the resulting expression.

This method is particularly handy because it provides a systematic approach to evaluating functions, making it easy even for those new to algebraic concepts.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that involves reducing an expression to its simplest form. In our example, after substituting \( x = 0 \) into the function \( f(x) = 3x - 7 \), we end up with \( 3(0) - 7 \).

• The first step in simplification is performing operations such as multiplication and subtraction.
• Multiplying \( 3 \) by \( 0 \) results in \( 0 \).
• Then, you subtract \( 7 \) to get \( -7 \).

The process of simplifying ensures that the solution is clear and concise, which is essential for accurately interpreting mathematical results.