Problem 43 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The values are \(h(-2) = -2\), \(h(-3) = -11\), \(h(4) = -32\), and \(h(5) = -51\).

Step 1: Identify the Function

We are given the function \( h(x) = -2x^2 - x + 4 \). This is a quadratic function that allows us to substitute different values of \(x\) to find corresponding values of \(h(x)\).

Step 2: Substitute and Calculate \(h(-2)\)

Substitute \(-2\) for \(x\) in the function: \( h(-2) = -2(-2)^2 - (-2) + 4 \).Calculate:(-2)^2 = 4, hence \( h(-2) = -2 \times 4 + 2 + 4 = -8 + 2 + 4 = -2 \).

Step 3: Substitute and Calculate \(h(-3)\)

Substitute \(-3\) for \(x\) in the function: \( h(-3) = -2(-3)^2 - (-3) + 4 \).Calculate: (-3)^2 = 9, hence \( h(-3) = -2 \times 9 + 3 + 4 = -18 + 3 + 4 = -11 \).

Step 4: Substitute and Calculate \(h(4)\)

Substitute \(4\) for \(x\) in the function: \( h(4) = -2(4)^2 - 4 + 4 \).Calculate: 4^2 = 16, hence \( h(4) = -2 \times 16 - 4 + 4 = -32 - 4 + 4 = -32 \).

Step 5: Substitute and Calculate \(h(5)\)

Substitute \(5\) for \(x\) in the function: \( h(5) = -2(5)^2 - 5 + 4 \).Calculate: 5^2 = 25, hence \( h(5) = -2 \times 25 - 5 + 4 = -50 - 5 + 4 = -51 \).

Key Concepts

Function Evaluation

Function evaluation is a core mathematical concept used to determine the output of a given function for specific inputs. In simplest terms, it involves plugging numbers into a function to see what the equation gives back.

Let's consider the function in this exercise, which is given by \[ h(x) = -2x^2 - x + 4. \]This is a quadratic function and we are interested in finding the values of this function when we substitute different numbers for the variable \(x\).

The process is simple:

• Start with specifying the value of \(x\) that you want to use. Example, \(x = -2\).
• Substitute this value into the function wherever \(x\) appears.
• Calculate the result to find \(h(x)\).

This gives us an easy way to evaluate complex functions by breaking them down into simpler arithmetic calculations.

Substitution Method

The substitution method is a fundamental technique for solving equations and problems involving functions. It involves replacing variables with specific values to simplify the function or to determine output values.

In the exercise, we used substitution to evaluate the function \(h(x) = -2x^2 - x + 4\) for different values of \(x\).

The steps are as follows:

• Choose the value that will be substituted for the variable \(x\).
• Carefully plug this value into every instance of \(x\) in the expression. For instance, replace \(x\) with \(-2\), \(-3\), \(4\), and \(5\) in separate instances.
• Perform the arithmetic operations according to mathematical rules (like order of operations).
• Get the result of the function with the substituted value.

This method is universal and can be applied across algebraic problems, making it an important tool for students to master.

Polynomial Functions

Polynomial functions are an essential topic in mathematics, characterized by expressions that comprise variables raised to whole number powers, with coefficients. In the given problem, we work with a quadratic polynomial function:\[ h(x) = -2x^2 - x + 4. \]This is because the highest power of \(x\) in the function is 2, making it a second-degree polynomial.

Polynomial functions, including quadratic ones, have certain properties:

• They have a finite degree determined by the highest power of the variable.
• The coefficients (like \(-2\) and \(-1\) here) determine the function's specific shape and behavior.
• The function can either have a maximum or minimum value depending on their degree.

Polynomial functions are pivotal in studying curves and patterns in mathematics. They also form the basis for more complex topics and are widely applied in real-world modeling and problem-solving.