Problem 45 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value of \(g(-3)\) is 30.

Step 1: Identify the function

We need to find the value of the function \(g(x) = 3x^2 - x\) at \(x = -3\).

Step 2: Substitute the value into the function

Substitute \(x = -3\) into the function \(g(x)\). This gives us \(g(-3) = 3(-3)^2 - (-3)\).

Step 3: Calculate the square

Calculate \((-3)^2\). The result is \(9\), so substitute back to get \(g(-3) = 3 \times 9 - (-3)\).

Step 4: Multiply and simplify

Multiply \(3\) and \(9\) to get \(27\). Now, modify the expression to \( g(-3) = 27 + 3 \).

Step 5: Add the terms

Add \(27\) and \(3\). The result is \(30\). Thus, \(g(-3) = 30\).

Key Concepts

Quadratic Function

A quadratic function is a type of function where the highest degree of the variable is two. The general form is given by \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratic functions create a curve called a parabola when graphed on a coordinate plane. This curve can either open upwards or downwards, depending on the sign of the coefficient \( a \). If \( a \) is positive, the parabola opens upwards. Conversely, if \( a \) is negative, it opens downwards. Quadratic functions are crucial in various fields like physics, engineering, and economics because they can describe natural phenomena such as projectile motion.

Function Evaluation

Function evaluation is the process of finding the output of a function for a particular input value. To evaluate a function at a specific point, you replace the variable with a given number and perform the necessary calculations. For example, in the exercise where we have \( g(x) = 3x^2 - x \), to find \( g(-3) \), we substitute \( x = -3 \) into the equation and simplify.

• Identify the function and variable
• Substitute the variable with the given value
• Perform calculations step by step to simplify

Evaluating functions carefully is essential as it helps in understanding how changes in input affect the output.

Substitution

Substitution is a method used in algebra where one replaces a variable in an equation with a given value or another expression. It's a fundamental technique for solving and simplifying expressions. In the example, the exercise requires substituting \( x = -3 \) into \( g(x) = 3x^2 - x \). By substituting, the expression becomes \( g(-3) = 3(-3)^2 - (-3) \). The goal of substitution is to replace variables with constants to solve or evaluate expressions. This technique is widely used not only in mathematics but also in sciences and computer programming, making it a versatile tool in any analytical work.

Polynomial Function

A polynomial function is an algebraic expression made up of terms called monomials. Each term has a coefficient, a variable, and a non-negative integer exponent. The general form of a polynomial function is \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where \( n \) is the degree of the polynomial, \( a_n, a_{n-1}, ..., a_1, a_0 \) are coefficients, and \( a_n eq 0 \).

In the given exercise, \( g(x) = 3x^2 - x \) is a polynomial function because it consists of two terms with constant coefficients and exponents, and the highest exponent is 2.

• Contains one or more terms
• Each term includes a constant and a variable raised to a power
• Can model various realistic scenarios

Polynomial functions are vital in mathematics because they are used to model real-life situations and solve complex problems.