Problem 24 - Algebra and Trigonometry

Question

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{lll}{\text { (a) }(f \circ f)(-1)} & {\text { (b) }(g \circ g)(2)}\end{array} $$

Step-by-Step Solution

Answer

\((f \circ f)(-1) = -29\) and \((g \circ g)(2) = -2\).

Step 1: Understanding Function Compositions

Function composition involves applying one function to the result of another function. When evaluating \((f \circ f)(x)\), it means applying function \(f\) to the result of \(f(x)\). Similarly, \((g \circ g)(x)\) involves applying function \(g\) to the result of \(g(x)\).

Step 2: Compute \(f(x)\) for \(x = -1\)

First, we need to calculate \(f(x)\) when \(x = -1\). Substitute \(-1\) into the function \(f(x) = 3x - 5\):\[f(-1) = 3(-1) - 5 = -3 - 5 = -8\]

Step 3: Evaluate \((f \circ f)(-1)\)

Next, use the result from Step 2 in the composition \(f(x)\). We need to compute \(f(-8)\):\[f(-8) = 3(-8) - 5 = -24 - 5 = -29\]

Step 4: Compute \(g(x)\) for \(x = 2\)

Now to evaluate \((g \circ g)(2)\), start by calculating \(g(x)\) for \(x = 2\). Substitute \(2\) into \(g(x) = 2 - x^2\):\[g(2) = 2 - (2)^2 = 2 - 4 = -2\]

Step 5: Evaluate \((g \circ g)(2)\)

Next, use the result from Step 4 in the composition \(g(x)\). Compute \(g(-2)\):\[g(-2) = 2 - (-2)^2 = 2 - 4 = -2\]

Step 6: State the Final Answers

From the previous steps, we've calculated that \((f \circ f)(-1) = -29\) and \((g \circ g)(2) = -2\).

Key Concepts

Evaluating Functions

When evaluating functions, you are essentially running a function using a particular input \(x\) to determine what the output or result will be. The function is a defined relationship, like an equation, that tells you exactly how to obtain the output when you know the input. In the example given, we have the function \(f(x) = 3x - 5\). To evaluate this for a specific value, such as \(x = -1\), you simply substitute \(-1\) into the function in place of \(x\).

• Substituting yields: \(f(-1) = 3(-1) - 5\).
• The resulting output is \(-8\), which means when \(-1\) is input into the function, the output is \(-8\).

This process is similar regardless of the function, which may be more complex algebraic expressions like \(g(x) = 2 - x^2\). Evaluating \(g(x)\) for \(x = 2\) involves substituting the value into the function:

• Calculation becomes \(g(2) = 2 - (2)^2 = -2\).

By evaluating functions, you can find out what values the function takes at specific arguments, which is crucial for further calculations like composition.

Algebraic Functions

Algebraic functions are expressions including variables like \(x\) that work according to algebraic rules. In our example, \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) are algebraic functions. They consist of constants, variables, and operations of addition, subtraction, multiplication, division, and powers.

• In \(f(x) = 3x - 5\), '3x' represents a linear transformation, and '-5' adjusts this output.
• In \(g(x) = 2 - x^2\), '2' is a constant and '\(-x^2\)' represents a quadratic transformation.

Understanding how to manipulate these expressions is key to both evaluating individual function values and performing advanced operations like function composition, where you use the output of one function as the input for another.
Using algebraic functions allows you to model a wide variety of mathematical scenarios and solve many different types of problems. They are the backbone of many areas of mathematics, providing a clear way to represent relationships between varying quantities.

Composite Functions

Composite functions involve combining two functions into one. This is what we mean with expressions like \((f \circ f)(x)\) or \((g \circ g)(x)\), pronounced "f composed with f" or "g composed with g." In a composite setup, you use the result of one function as the input for another.

• For \((f \circ f)(-1)\), first evaluate \(f(x)\) at \(x = -1\), giving \(-8\).
• Then, use this result to evaluate \(f(x)\) again: \(f(-8) = -29\).

Composite functions are useful in scenarios where processes happen in sequence, and each stage depends on the previous one. They are particularly valuable in applied mathematics and various engineering fields. Understanding how to compute and utilize composites extends your ability to relate multiple stages of transformations and calculations.