Problem 74 - College Algebra

Question

For the following exercises, evaluate the expressions, given functions \(f, g\), and \(h\) : $$ f(x)=3 x-2 \quad g(x)=5-x^{2} \quad h(x)=-2 x^{2}+3 x-1 $$ $$ 3 f(1)-4 g(-2) $$

Step-by-Step Solution

Answer

The value of the expression is -1.

Step 1: Evaluate f(1)

Plug in the value of 1 for the variable x in the function \( f(x) = 3x - 2 \). Compute \( f(1) = 3(1) - 2 \) which simplifies to \( 1 \).

Step 2: Evaluate g(-2)

Plug in the value of -2 for the variable x in the function \( g(x) = 5 - x^2 \). Compute \( g(-2) = 5 - (-2)^2 \) which simplifies to \( 1 \).

Step 3: Multiply by Coefficients

Substitute the values from Steps 1 and 2 into the expression \( 3f(1) - 4g(-2) \). Here \( 3f(1) = 3(1) \) and \( 4g(-2) = 4(1) \).

Step 4: Perform the Arithmetic

Evaluate \( 3f(1) - 4g(-2) \) by computing \( 3 \times 1 - 4 \times 1 = 3 - 4 = -1 \).

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value into a function to find the output. In this exercise, we have three functions: \( f(x) = 3x - 2 \), \( g(x) = 5 - x^2 \), and \( h(x) = -2x^2 + 3x - 1 \). To evaluate a function, you replace the variable \( x \) with the value given and then perform the necessary arithmetic operations.

For example, in Step 1 of the solution, to evaluate \( f(1) \), you substitute 1 into the function \( f \):

• Calculate \( f(1) = 3(1) - 2 \).
• This results in \( f(1) = 1 \).

This process allows us to see what output the function produces for the input of \( x = 1 \). Function evaluation allows us to explore how functions behave at specific points, critical in understanding the properties of algebraic functions.

Polynomial Functions

Polynomial functions, like those in the exercise, consist of terms composed of variables raised to whole number powers, multiplied by coefficients, and summed together. They are foundational in algebra and help describe a variety of real-world phenomena.

Each term in a polynomial function is of the form \( ax^n \), where \( a \) is a constant coefficient, and \( n \) is a non-negative integer. The highest power of \( x \) in the function determines its degree. For instance:

• For \( f(x) = 3x - 2 \), the degree is 1, making it a linear polynomial.
• For \( g(x) = 5 - x^2 \), the degree is 2, making it a quadratic polynomial.

Polynomial functions are versatile, allowing both simple shifts and complex curves, accommodating various mathematical operations. Understanding them is key to mastering more advanced algebraic concepts.

Basic Arithmetic Operations

Basic arithmetic operations form the basis of evaluating algebraic functions and include addition, subtraction, multiplication, and division. These operations are used step-by-step in the solution of the given expression.

Let's look at the expression \( 3f(1) - 4g(-2) \):

• First, we evaluate \( f(1) \) and \( g(-2) \) as provided by substituting specific values into the functions.
• Then, we multiply these evaluated results by their respective coefficients, 3 for \( f(1) \) and 4 for \( g(-2) \).

Finally, using subtraction, we combine these results:
- Multiply: \( 3 \times 1 = 3 \) and \( 4 \times 1 = 4 \).
- Subtract: \( 3 - 4 = -1 \).

Understanding these operations and how to execute them correctly ensures accuracy in problem-solving and builds a strong foundation for tackling complex algebraic problems.