Problem 97 - Algebra for College Students

Question

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(2)+g(3) $$

Step-by-Step Solution

Answer

The result is 2.

Step 1 - Evaluate \( f(2) \)

First, find the value of \( f(x) \) when \( x = 2 \). Use the given function \( f(x) = 3x - 2 \):\[ f(2) = 3(2) - 2 \]Solve for \( f(2) \):\[ f(2) = 6 - 2 = 4 \]

Step 2 - Evaluate \( g(3) \)

Next, find the value of \( g(x) \) when \( x = 3 \). Use the given function \( g(x) = -x^2 + 3x - 2 \):\[ g(3) = -(3)^2 + 3(3) - 2 \]Solve for \( g(3) \):\[ g(3) = -9 + 9 - 2 = -2 \]

Step 3 - Add \( f(2) \) and \( g(3) \)

Now add the values found in the previous steps. We have \( f(2) = 4 \) and \( g(3) = -2 \):\[ f(2) + g(3) = 4 + (-2) = 2 \]

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics. It involves finding the value of a function for a particular input. Consider a function as a special machine that takes an input (x) and gives an output (f(x)). For example, if we have a function \( f(x) = 3x - 2 \), evaluating this function for \( x = 2 \) means calculating the value of f(2). We simply replace every \( x \) in the function with 2 and perform the arithmetic operations.

This concept is crucial because it builds the foundation for more complex topics in mathematics such as calculus, algebra, and even in applied fields like physics and engineering. Always remember the steps:

• Identify the function and the input value.
• Substitute the input value in place of the variable.
• Perform the necessary calculations.

Understanding how to evaluate a function helps in solving various kinds of mathematical problems efficiently.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents on the variables. For instance, \( g(x) = -x^2 + 3x - 2 \) is a polynomial function.

These functions are very versatile and appear frequently in different areas of mathematics and its applications. Here are some common characteristics:

• They consist of terms with variables raised to whole-number exponents.
• The highest power of the variable is known as the degree of the polynomial (In the example above, the degree is 2).
• Polynomial functions can have one or multiple terms (monomial, binomial, trinomial, etc.).

To evaluate a polynomial function like \( g(3) \), replace every instance of the variable with the given value and solve the resulting arithmetic expression. This results in understanding how changes in the input affect the output, a key aspect in graphing and analyzing functions.

Absolute Value Function

The absolute value function, denoted by \( |x| \), measures the distance of a number from zero on the number line, regardless of direction. Essentially, it turns any negative value positive while keeping positive values unchanged.

For the function \( h(x) = |x + 2| \), the absolute value impacts the behavior significantly:

• If \( x + 2 \) is positive or zero, \( h(x) = x + 2 \).
• If \( x + 2 \) is negative, \( h(x) = -(x + 2) \).

This results in a V-shaped graph when plotted, where the vertex (the point where the function changes direction) is at the point where the expression inside the absolute value equals zero.

Understanding absolute value functions helps solve a variety of problems, especially those involving measurements and distances. Practicing the evaluation of functions like \( h(x) \) when \( x \) is within different ranges strengthens the grasp of interpreting and working with these functions in practical scenarios.