Problem 41 - Intermediate Algebra

Question

If \(g(x)=2 x^{2}-5 x-7\), find \(g(-1), g(2), g(-3)\), and \(g(4)\).

Step-by-Step Solution

Answer

0, -9, 26, 5

Step 1: Substitute x with -1 in g(x)

To find \(g(-1)\), substitute \(x = -1\) into the function \(g(x) = 2x^2 - 5x - 7\).\\[g(-1) = 2(-1)^2 - 5(-1) - 7\] \Simplify the expression:\[g(-1) = 2(1) + 5 - 7 = 2 + 5 - 7\] \Calculate: \\[g(-1) = 0\]

Step 2: Substitute x with 2 in g(x)

To find \(g(2)\), substitute \(x = 2\) into the function \(g(x) = 2x^2 - 5x - 7\).\\[g(2) = 2(2)^2 - 5(2) - 7\] \Simplify the expression: \\[g(2) = 2(4) - 10 - 7 = 8 - 10 - 7\] \Calculate: \\[g(2) = -9\]

Step 3: Substitute x with -3 in g(x)

To find \(g(-3)\), substitute \(x = -3\) into the function \(g(x) = 2x^2 - 5x - 7\).\\[g(-3) = 2(-3)^2 - 5(-3) - 7\] \Simplify the expression: \\[g(-3) = 2(9) + 15 - 7 = 18 + 15 - 7\] \Calculate: \\[g(-3) = 26\]

Step 4: Substitute x with 4 in g(x)

To find \(g(4)\), substitute \(x = 4\) into the function \(g(x) = 2x^2 - 5x - 7\).\\[g(4) = 2(4)^2 - 5(4) - 7\] \Simplify the expression: \\[g(4) = 2(16) - 20 - 7 = 32 - 20 - 7\] \Calculate: \\[g(4) = 5\]

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases constructed from variables, coefficients, and arithmetic operations like addition, subtraction, multiplication, and non-negative integer exponents. In our exercise, the function \( g(x) = 2x^2 - 5x - 7 \) is a quadratic polynomial expression. The highest power of the variable \( x \) is 2, indicating it is a quadratic function. This type of polynomial is recognized by its standard form: \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero.
Polynomial expressions are significant because they model a wide range of real-world situations ranging from physics to economics. Understanding how to manipulate and evaluate them is integral to solving equations and understanding the behavior of functions. Quadratic functions, in particular, often have parabolic graphs and can model projectile motion, optimize problems, and more.

Substitution Method

The substitution method is a useful technique for finding the value of a function at a particular point. It involves replacing the variable in a function with a specific numerical value. In our exercise, we've used substitution to find \( g(-1) \), \( g(2) \), \( g(-3) \), and \( g(4) \).
Here's how it works:

• Take the original function: \( g(x) = 2x^2 - 5x - 7 \).
• Replace \( x \) with the desired value, such as \( -1 \), \( 2 \), \( -3 \), or \( 4 \).
• Perform the arithmetic operations to simplify the expression.
• This provides the function's output for that specific input.

The power of this method lies in its simplicity and direct application to solve numerous algebraic problems quickly. It is particularly useful in functional evaluation, where you need precise output without solving for general expressions.

Function Evaluation

Function evaluation is the process of finding the output of a function for particular input values. When you substitute the input values into the function and simplify, you evaluate the function. In our exercise with \( g(x) = 2x^2 - 5x - 7 \), evaluating means finding what \( g(x) \) equals when \( x \) is \(-1\), \(2\), \(-3\), and \(4\).
Important points to remember about function evaluation:

• It involves substituting numbers into the function's formula.
• Function evaluation helps understand the pattern and behavior of a function at various points.
• Once substituted, simplify the expression to find the final output.
• It’s a primary tool in understanding how functions model and are analyzed in practical scenarios.

Through this process, function evaluation aids in predicting trends, interpreting real-world data, and analyzing systems subject to change.