Problem 30 - Precalculus : Building Concepts and Connections

Question

In Exercises \(17-40,\) let \(f(x)=-x^{2}+x, g(x)=\frac{2}{x+1},\) and \(h(x)=-2 x+1 .\) Evaluate each of the following. $$(f g)(-3)$$

Step-by-Step Solution

Answer

The value of \((f g)(-3)\) is 12.

Step 1: Identify the functions

The given functions are \(f(x)=-x^{2}+x, g(x)=\frac{2}{x+1}\) and our task is to evaluate \((f g)(-3)\), which is equivalent to \(f(-3)g(-3)\).

Step 2: Calculate \(f(-3)\)

To calculate \(f(-3)\), replace the \(x\) values in \(f(x)\) with -3: \(-(-3)^{2}+(-3)=-9-3=-12.\) So, \(f(-3)=-12\)

Step 3: Calculate \(g(-3)\)

Similarly, replace the \(x\) values in \(g(x)\) with -3: \(\frac{2}{-3+1}=\frac{2}{-2}=-1\). So, \(g(-3)=-1\)

Step 4: Evaluate \((f g)(-3)\)

Finally, to calculate \((f g)(-3)\), multiply \(f(-3)\) by \(g(-3)\): \(-12*(-1)=12\).

Step 5: Conclusion

Hence, the product \((f g)(-3)\) equals 12.

Key Concepts

Polynomial Functions

Polynomial functions are one of the foundational building blocks in algebra and calculus. They are expressions that consist of variables and coefficients combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, in the function given as part of the exercise, \( f(x) = -x^2 + x \), it's a polynomial of degree 2 because the highest power of \( x \) is 2.

Key characteristics of polynomial functions include:

• Degree: The highest exponent of the variable \( x \) determines the degree of the polynomial, which influences its graph's shape.
• Coefficients: These are the numbers in front of the variables, such as \(-1\) and \(1\) in the given \( f(x) \).
• Constant Term: Polynomials may have a term without a variable, although \( f(x) \) doesn't have one.

Polynomials can include terms like \( x^3 \), \( x^2 \), \(2x\), or constants like 5. Different operations on polynomials including addition and multiplication, modify their degrees and coefficients.
Understanding these basic properties allows for easy evaluation and simplification of polynomial expressions during problem-solving.

Rational Functions

Rational functions are another important category of functions in mathematics. They can be thought of as the ratio of two polynomial functions. Mathematically, a rational function is represented as \( R(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \).

In the exercise problem, \( g(x) = \frac{2}{x+1} \) is a rational function. Here, \( P(x) = 2 \) and \( Q(x) = x+1 \). It's essential to understand the domain of rational functions, which includes all real numbers except where \( Q(x) = 0 \). In this case, \( g(x) \) is undefined when \( x = -1 \).

• Discontinuities: These occur where the denominator \( Q(x) \) is zero.
• Asymptotes: Vertical asymptotes are common, occurring at points of discontinuity, where the function rapidly increases or decreases towards infinity.

Understanding rational functions helps in modeling situations where quantities are divided by varying numbers, which is common in real-world situations such as rates and proportions.

Composite Functions

Composite functions are created when two functions are combined together, using the output of one function as the input of another. This concept is frequently utilized in function evaluation and transformation. A composite function is expressed as \( (f \circ g)(x) = f(g(x)) \), which signifies that you apply \( g \) first and then apply \( f \) to the result. In the exercise, solving \( (f g)(-3) \) involves multiplying the results of \( f(-3) \) and \( g(-3) \).

Important aspects of composite functions include:

• Order of Operations: It matters in composite functions because you must find the inner function value first before substituting into the outer function.
• Domains: The domain of the composite function \( f(g(x)) \) is dependent on the domains of both \( f \) and \( g \), and any restrictions that may apply from their compositions.

Composite functions allow for the representation of complex processes and transformations, making them useful in various applications in mathematics and beyond. By understanding and manipulating these functions, we can model complicated systems in a more simplified manner.