Problem 75 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the expressions, given functions \(f, g,\) and \(h:\) \(f(x)=3 x-2\) \(g(x)=5-x^{2}\) \(h(x)=-2 x^{2}+3 x-1\) \(f\left(\frac{7}{3}\right)-h(-2)\)

Step-by-Step Solution

Answer

The evaluated expression is 20.

Step 1: Substitute into f(x)

Replace the variable in the function \( f(x) = 3x - 2 \) with \( x = \frac{7}{3} \). This gives: \( f\left(\frac{7}{3}\right) = 3\left(\frac{7}{3}\right) - 2 \).

Step 2: Calculate f(x)

Simplify the expression from the previous step: \( 3\left(\frac{7}{3}\right) - 2 = 7 - 2 = 5 \).

Step 3: Substitute into h(x)

Now replace \( x = -2 \) in the function \( h(x) = -2x^2 + 3x - 1 \). This results in: \( h(-2) = -2(-2)^2 + 3(-2) - 1 \).

Step 4: Calculate h(x)

Simplify \( h(-2) = -2(4) + (-6) - 1 = -8 - 6 - 1 = -15 \).

Step 5: Evaluate the expression

Subtract the result of \( h(-2) \) from \( f\left(\frac{7}{3}\right) \): \( f\left(\frac{7}{3}\right) - h(-2) = 5 - (-15) \).

Step 6: Simplify the final expression

Evaluate the expression further: \( 5 - (-15) = 5 + 15 = 20 \).

Key Concepts

Algebraic Functions

Algebraic functions are a fundamental concept in mathematics, involving expressions composed of variables and constants connected through operations like addition, subtraction, multiplication, and division. They can also include powers and roots. Understanding how to work with these functions is crucial as they appear frequently in various mathematical problems.
Algebraic functions are typically expressed in the form of equations, such as \( f(x) = 3x - 2 \) or \( h(x) = -2x^2 + 3x - 1 \). These functions make it easy to determine the output based on a given input (the variable, usually represented by \( x \)).

• **Linear Function Example: ** The function \( f(x) = 3x - 2 \) is linear because it forms a straight line when graphed and has no variables raised to powers higher than one.
• **Quadratic Function Example: ** The function \( h(x) = -2x^2 + 3x - 1 \) is a quadratic since it includes the term \( -2x^2 \), indicating a parabola in its graph.

Algebraic functions serve as a bridge to more advanced topics in mathematics, providing a clear and logical way to model relationships.

Substitution Method

The substitution method is a technique used to evaluate expressions, making it effective to find specific values of functions by replacing variables with numbers.
This method involves directly substituting the given value into the function and simplifying the resulting expression. It is a straightforward approach that helps simplify complex expressions by breaking them down into manageable parts.

For example, in the function \( f(x) = 3x - 2 \), substituting \( x = \frac{7}{3} \), involves:

• Replacing \( x \) with \( \frac{7}{3} \): \( f\left(\frac{7}{3}\right) = 3\left(\frac{7}{3}\right) - 2 \).
• Simplifying to find \( f\left(\frac{7}{3}\right) = 7 - 2 = 5 \).

Similarly, when substituting \( x = -2 \) into \( h(x) = -2x^2 + 3x - 1 \):

• The substitution results in \( h(-2) = -2(-2)^2 + 3(-2) - 1 \).
• Further simplification gives \( h(-2) = -8 - 6 - 1 = -15 \).

This method not only simplifies the computation but also allows a clearer understanding of how changes in the input affects the output.

Expression Simplification

Expression simplification involves reducing mathematical expressions to their simplest form while maintaining their original value. This process is essential for making complex problems easier to solve.
It often includes combining like terms, performing arithmetic operations, and removing any unnecessary parts of the expression.

Take for instance, the final calculation in our example:

• After substituting and simplifying \( f\left(\frac{7}{3}\right) \) and \( h(-2) \), we get results 5 and -15, respectively.
• To simplify the full expression \( f\left(\frac{7}{3}\right) - h(-2) \), the computation involves subtracting \(-15\) from 5.
• The simplification process yields \( 5 - (-15) \), which further reduces to \( 5 + 15 \) or 20.

Expression simplification helps in revealing the intrinsic value of the expression, making it easier to interpret and use in different contexts. By simplifying expressions, you can more clearly see the relationships between variables, which aids in building mathematical intuition.