Problem 55 - Intermediate Algebra

Question

If \(f(x)=5 x^{2}-2 x+3\) and \(g(x)=-x^{2}+4 x-5\), find \(f(-2), f(3), g(-4)\), and \(g(6)\).

Step-by-Step Solution

Answer

The evaluated values are: \(f(-2) = 27\), \(f(3) = 42\), \(g(-4) = -37\), and \(g(6) = -17\).

Step 1: Evaluate f(-2)

Substitute \(-2\) for \(x\) in the function \(f(x) = 5x^2 - 2x + 3\). Calculate: \[f(-2) = 5(-2)^2 - 2(-2) + 3 = 5(4) + 4 + 3 = 20 + 4 + 3 = 27\] Therefore, \(f(-2) = 27\).

Step 2: Evaluate f(3)

Substitute \(3\) for \(x\) in the function \(f(x) = 5x^2 - 2x + 3\). Calculate: \[f(3) = 5(3)^2 - 2(3) + 3 = 5(9) - 6 + 3 = 45 - 6 + 3 = 42\] Therefore, \(f(3) = 42\).

Step 3: Evaluate g(-4)

Substitute \(-4\) for \(x\) in the function \(g(x) = -x^2 + 4x - 5\). Calculate: \[g(-4) = -(-4)^2 + 4(-4) - 5 = -(16) - 16 - 5 = -16 - 16 - 5 = -37\] Therefore, \(g(-4) = -37\).

Step 4: Evaluate g(6)

Substitute \(6\) for \(x\) in the function \(g(x) = -x^2 + 4x - 5\). Calculate: \[g(6) = -(6)^2 + 4(6) - 5 = -36 + 24 - 5 = -36 + 24 - 5 = -17\] Therefore, \(g(6) = -17\).

Key Concepts

Substitution in Equations

Substitution is a key technique in algebra, especially for evaluating functions. The goal of substitution is to replace a variable with a number or another expression. It allows us to find the value of a function for a specific input. In the context of evaluating functions like \( f(x) = 5x^2 - 2x + 3 \), we substitute different values of \( x \) to see how the function behaves for those specific cases.

• Identify the variable to substitute, which is often \( x \) in these problems.
• Replace \( x \) with the given number in the function equation.
• Carefully follow the order of operations: evaluate exponents first, followed by multiplication and division, and then addition and subtraction.

For example, to find \( f(-2) \), we replace \( x \) with \(-2\) and simplify the expression step by step: \[f(-2) = 5(-2)^2 - 2(-2) + 3\] Simplifying it involves squaring the \(-2\), multiplying, and then adding or subtracting the constants. This leads us to a final value of 27 for the example given.

Quadratic Functions

Quadratic functions, like \( f(x) = 5x^2 - 2x + 3 \) and \( g(x) = -x^2 + 4x - 5 \), are polynomial functions of degree two. These functions generally plot as parabolas on a graph. Here are some characteristics:

• The general form is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.
• The coefficient \( a \) determines the direction of the parabola (upward if \( a > 0 \), downward if \( a < 0 \)).
• The vertex is the turning point of the parabola and can either represent a maximum or a minimum value.

Understanding these features helps in determining how the function values change as \( x \) changes. By evaluating specific points, such as \( f(-2) \) or \( g(6) \), we get numeric insights into the function's behavior at those points.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. When evaluating functions, such as \( f(x) = 5x^2 - 2x + 3 \), it's essential to handle each piece of the expression correctly according to algebraic rules.

• Use parentheses to clarify operations and ensure correct order of precedence.
• Handle each term separately. For \( f(x) \), split it into three parts: \( 5x^2 \), \(-2x\), and \( +3 \).
• Apply the basic operations: multiplication and exponentiation are typically done first, followed by the addition or subtraction of terms.

Algebraic expressions form the backbone of much of algebra, serving as the foundation for more complex equations and functions. By practicing with simple expressions, students gain confidence and skills needed to tackle harder problems with ease.