Problem 44 - Algebra 2

Question

Find each value if \(f(x)=6 x+2\) and \(g(x)=3 x^{2}-x\). \(g(1)\)

Step-by-Step Solution

Answer

The value of \( g(1) \) is 2.

Step 1: Identify the Function

The function given for this exercise is \( g(x) = 3x^2 - x \). We need this function to find \( g(1) \).

Step 2: Substitute the Value into the Function

Substitute the value \( x = 1 \) into the function \( g(x) = 3x^2 - x \). So, calculate \( g(1) = 3(1)^2 - 1 \).

Step 3: Calculate the Expression

First, compute \( 1^2 \), which is \( 1 \). Then multiply by 3 to get \( 3 \times 1 = 3 \).

Step 4: Simplify the Function Evaluation

Subtract the value of \( x = 1 \) from \( 3 \). Therefore, \( 3 - 1 = 2 \).

Key Concepts

Substitution in Algebra

Substitution in algebra involves replacing a variable in an expression with a specific value to simplify the expression or solve an equation. This process is fundamental in solving function evaluations. For instance, if we have a function \( g(x) = 3x^2 - x \) and we need to find \( g(1) \), we're using substitution.
The steps involved are simple:

• Identify the variable in the expression. In our function, \( x \) is the variable.
• Replace the variable with the given number. Here, we replace \( x \) with 1.
• Recalculate using the substituted value to find the result.

By substituting \( x = 1 \) into \( g(x) \), we perform calculations step-by-step, ensuring accuracy, and finally arrive at a numerical answer. This method allows us to solve for specific outputs where the input is known.

Polynomial Functions

Polynomial functions are algebraic expressions involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, these are expressions made up of terms like \( ax^n \), where \( a \) is a coefficient, \( x \) is the variable, and \( n \) is a non-negative integer called the degree of the term.
Our function \( g(x) = 3x^2 - x \) is a polynomial. It consists of:

• A quadratic term \( 3x^2 \), where the degree of the term is 2 because of the exponent on \( x \).
• A linear term \( -x \), as it has a degree of 1.

Polynomial functions are versatile and fundamental in algebra, allowing us to model a wide range of scenarios and behaviors. They help in understanding trends in data and predicting future outcomes by analyzing the relation between variables.

Quadratic Expressions

Quadratic expressions are a specific type of polynomial where the highest exponent of the variable is 2. These expressions are typically of the form \( ax^2 + bx + c \), with \( a \), \( b \), and \( c \) as constants and \( a eq 0 \).
In the function \( g(x) = 3x^2 - x \), the part \( 3x^2 \) represents the quadratic term. Quadratic expressions usually form a U-shaped curve called a parabola when graphed. Some characteristics of quadratics are:

• The parabola can open upward or downward, depending on the sign of \( a \). Here, since \( a = 3 \), it opens upwards.
• The vertex, which is the turning point of the parabola, provides critical information about the maximum or minimum value of the function.

Quadratics are used extensively in solving real-world problems involving area calculations, projectile motions, and economics.