Problem 77 - College Algebra with Modeling and Visualization

Question

Exercises 77 and 78: Numerical representations for the functions \(f\) and \(g\) are given. Evaluate the expression, if possible. $$ \begin{array}{llll} \text { (a) }(g \circ f)(1) & \text { (b) }(f \circ g)(4) & \text { (c) }(f \circ f)(3) \end{array} $$ $$ \begin{array}{rrrrr} x & 1 & 2 & 3 & 4 \\ f(x) & 4 & 3 & 1 & 2 \end{array} $$ $$ \begin{array}{rrrrr} x & 1 & 2 & 3 & 4 \\ g(x) & 2 & 3 & 4 & 5 \end{array} $$

Step-by-Step Solution

Answer

(a) 5; (b) Unable to evaluate; (c) 4.

Step 1: Evaluate \((g \circ f)(1)\)

The expression \((g \circ f)(1)\) means we first find \(f(1)\) and then use that result for the function \(g\). From the given table for \(f(x)\), we have \(f(1) = 4\). Next, we find \(g(4)\) using the table for \(g(x)\), which gives us \(g(4) = 5\). Therefore, \((g \circ f)(1) = 5\).

Step 2: Evaluate \((f \circ g)(4)\)

The expression \((f \circ g)(4)\) means we first find \(g(4)\) and then use that result for the function \(f\). From the table for \(g(x)\), \(g(4) = 5\). Now, we look for \(f(5)\) in the table for \(f(x)\), but there is no value listed for \(x = 5\). Therefore, \((f \circ g)(4)\) cannot be evaluated with the given tables.

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

The expression \((f \circ f)(3)\) means that we first apply \(f o(3)\), then apply \(f\) again to the result. From the table for \(f(x)\), we know \(f(3) = 1\). Now, we need \(f(1)\), which is \(f(1) = 4\) according to the table. Therefore, \((f \circ f)(3) = 4\).

Key Concepts

Numerical Representation

In the study of functions, having a clear understanding of the numerical representation is crucial. Numerical representation refers to the way in which functions are depicted using tables or charts.
These tables provide concrete values of a function for given inputs.
For example, in our exercise, functions \(f\) and \(g\) are presented in tables that list outputs for specific inputs.

• The table for \(f\) shows the outputs \([4, 3, 1, 2]\) corresponding to inputs \([1, 2, 3, 4]\).
• The table for \(g\) lists outputs \([2, 3, 4, 5]\) for the same inputs \([1, 2, 3, 4]\).

These tables make it easy to see exactly what value to plug into a function for a particular input. They enhance our understanding of the functions by providing a straightforward numerical perspective.
Being able to read and interpret these representations is a fundamental skill in mathematics, as it allows for precise function evaluations.

Function Evaluation

Function evaluation involves the process of determining the output of a function for a specific input.
This is done by looking up the corresponding value in the function's numerical representation. For example, if you're asked to evaluate \(f(1)\), you would simply find \(1\) in the \(x\) row of the \(f(x)\) table and locate the corresponding output, \(4\) in this case.
This straightforward process is a crucial part of working with functions.Function evaluation is essential because it allows us to see the behavior of the function at specific points.
In the step-by-step solution, we saw how \(f(1) = 4\) and \(g(4) = 5\), meaning for these inputs, the evaluated outputs are the respective numbers.
This basic idea extends to more complex operations, such as composing multiple functions, where the output of one function becomes the input of another.

Composite Function

Composite functions bring about an interesting dynamic in mathematics.
It's when two functions combine in such a way that the output of one function becomes the input of another.
For representation, we write it as \((g \circ f)(x)\) or \((f \circ g)(x)\).To evaluate a composite function, the key is performing function evaluations in sequence.
Let's consider the expression \((g \circ f)(1)\).

• Start by finding \(f(1)\), which equals \(4\).
• Then, use this result as the input in the function \(g\), so you find \(g(4)\), equaling \(5\).

Thus, \((g \circ f)(1) = 5\). Composite functions enable us to explore more intricate relationships between variables and outcomes.
They highlight the importance of understanding each function individually before evaluating how they interact when combined. By mastering this concept, you can confidently tackle complex problems involving multiple functions.