Problem 48 - Algebra A Combined Function

Question

If \(f(x)=3 x+3, g(x)=4 x^{2}-6 x+3,\) and \(h(x)=5 x^{2}-7,\) find each function value. \(f(-1)\)

Step-by-Step Solution

Answer

The value of \(f(-1)\) is 0.

Step 1: Understanding the Function

The function given is \(f(x) = 3x + 3\). This is a linear function with a slope of 3 and a y-intercept of 3.

Step 2: Substitute the Value of x

To find \(f(-1)\), substitute \(-1\) for \(x\) in the function \(f(x)\). This gives us \(f(-1) = 3(-1) + 3\).

Step 3: Perform the Multiplication

Calculate the multiplication part of the expression: \(3(-1)\). This results in \(-3\).

Step 4: Perform the Addition

Add the result from Step 3 to the constant value in the function: \(-3 + 3\). This results in 0.

Key Concepts

Linear Function

A linear function is a type of function that creates a straight line when graphed. It has the general form of \(f(x) = ax + b\), where \(a\) and \(b\) are constants. In our exercise, the function \(f(x) = 3x + 3\) fits this form, where \(a = 3\) is the slope and \(b = 3\) is the y-intercept.

• The slope \(a = 3\) tells us how steep the line is; it shows the change in \(y\) for a one-unit change in \(x\).
• The y-intercept \(b = 3\) indicates where the line crosses the y-axis.

These characteristics make it easy to graph or to predict outputs for given inputs. Linear functions are common in various practical fields like economics and physics, where they are used to model relationships with constant rates of change.

Substitution Method

The substitution method is a straightforward technique used in algebra to find the output of a function for a given input. To apply this method, one needs to replace each occurrence of the variable \(x\) with the specific number provided. In this exercise, you substitute \(-1\) into \(f(x) = 3x + 3\). This action changes the function to \(f(-1) = 3(-1) + 3\).

• It's a reliable way to evaluate functions quickly.
• Allowed us to check our mathematical model and methodological accuracy.
• It's particularly useful for solving equations or checking solutions.

Understanding substitution is essential as it lays the foundation for more complex algebraic manipulations and solving systems of equations.

Arithmetic Operations

Arithmetic operations involve basic calculations, such as addition, subtraction, multiplication, and division, which are foundational in mathematics. In the given solution:1. **Multiplication**: We started with the expression \(3(-1)\). Here, multiplication involves changing the sign and magnitude of numbers. 2. **Addition**: Next, the operation -3 + 3 was performed. Addition can either accumulate values or, when combining opposition numbers like here, neutralize them to zero. Performing arithmetic operations correctly is crucial in determining precise evaluation of functions and ensuring accurate results.

• Mastering these skills leads to higher-level problem-solving.
• They are used daily in numerous mathematical practices.

Grasping these basic operations helps students perform not only academic tasks with confidence but also everyday decisions involving numbers.