Problem 90 - Algebra 2

Question

If \(f(x)=3-x\) and \(g(x)=x^{2}-3,\) which expression has the greatest value? $$\begin{array}{llll}{\text { F. }(g \circ f)(-3)} & {\text { G. }(f \circ g)(-3)} & {\text { H. }(f \cdot g)(-3)} & {\text { J. }(g-f)(-3)}\end{array}$$

Step-by-Step Solution

Answer

Expression H \((f \cdot g)(-3)\) has the greatest value, \(36\).

Step 1: Compute Expression F

Calculate \((g\circ f)(-3)\). This means we plug \(f(-3)\) into \(g(x)\). Because \(f(-3) = 3 - (-3) = 6\), the expression becomes \(g(6) = 6^2 - 3 = 36 - 3 = 33\).

Step 2: Compute Expression G

Calculate \((f\circ g)(-3)\). This means we plug \(g(-3)\) into \(f(x)\). Because \(g(-3) = (-3)^2 - 3 = 9 - 3 = 6\), the expression becomes \(f(6) = 3 - 6 = -3\).

Step 3: Compute Expression H

Calculate \((f \cdot g)(-3)\). This means we multiply \(f(-3)\) by \(g(-3)\), so the expression becomes \(f(-3) \cdot g(-3) = 6 \cdot 6 = 36\).

Step 4: Compute Expression J

Calculate \((g-f)(-3)\). This means we subtract \(f(-3)\) from \(g(-3)\). So the expression becomes \(g(-3) - f(-3) = 6 - 6 = 0\).

Step 5: Compare Values

Now, all that's left is to compare the values. We find that \(36 > 33 > 0 > -3\). So the expression that has the greatest value is expression H, \((f \cdot g)(-3) = 36\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function for a specific input. Essentially, it involves substituting a given number in place of the function's variable, usually denoted by \( x \). For example, to evaluate \( f(x) = 3 - x \) at \( x = -3 \), we replace \( x \) with \( -3 \) to get \( f(-3) = 3 - (-3) = 6 \).
Function evaluation is crucial in determining outputs that help in graphing functions or analyzing their behavior. In composition problems, you evaluate multiple functions in a sequence. If you need to evaluate \((g \circ f)(x)\), you first find \( f(x) \) and then use its value to evaluate \( g \). By mastering function evaluation, you simplify complex operations involving multiple functions.

Polynomials

Polynomials are mathematical expressions that involve variables raised to whole number powers and their coefficients. They appear as sums or differences of terms like \( x^2 \), \( 3x \), or constant terms like \( -5 \).
A polynomial can have one or many terms and can be of any degree, depending on the highest power of the variable. For instance, \( g(x) = x^2 - 3 \) is a simple polynomial of degree two, known as a quadratic polynomial.
Polynomials are foundational in algebra because they provide a framework for expressing mathematical relationships and solving equations. When you analyze functions like \( g(x) \) or multiply them, like in the function composition tasks, understanding polynomials becomes essential. This understanding helps you to manipulate terms correctly and find maximum or minimum values effectively.

Order of Operations

The order of operations is a rule that helps us determine which calculations to perform first in a complex mathematical expression. This principle is commonly remembered by the acronym PEMDAS/BODMAS:

• P/B: Parentheses/Brackets
• E/O: Exponents/Orders (such as squares or square roots)
• MD/DM: Multiplication and Division (left to right)
• AS: Addition and Subtraction (left to right)

In function composition or any polynomial operation, following the order of operations is crucial. For example, in the expression \( (g \circ f)(-3) \), we first find \( f(-3) \), and then substitute that value into \( g \), following each step carefully ensures accuracy.
By adhering strictly to these rules, you prevent miscalculations that could lead to incorrect results. Consistently applying the order of operations lets anyone tackle math problems with confidence and clarity.