Question

Given \(f(2)=3, f(3)=4, f(5)=0, g(2)=5\), \(g(3)=2,\) and \(g(4)=-1,\) evaluate the following. a) \(f(g(3))\) b) \(f(g(2))\) c) \(g(f(2))\) d) \(g(f(3))\)

Short answer

a) 3, b) 0, c) 2, d) -1

Step-by-step solution

- Understand Given Data

Review the values given for the functions: \(f(2)=3\), \(f(3)=4\), \(f(5)=0\), \(g(2)=5\), \(g(3)=2\), \(g(4)=-1\).

- Evaluate \(f(g(3))\)

First, find the value of \(g(3)\), which is 2. Then evaluate \(f(2)\). Given \(f(2)=3\), we have: \(f(g(3)) = f(2) = 3\).

- Evaluate \(f(g(2))\)

First, find the value of \(g(2)\), which is 5. Then evaluate \(f(5)\). Given \(f(5)=0\), we have: \(f(g(2)) = f(5) = 0\).

- Evaluate \(g(f(2))\)

First, find the value of \(f(2)\), which is 3. Then evaluate \(g(3)\). Given \(g(3)=2\), we have: \(g(f(2)) = g(3) = 2\).

- Evaluate \(g(f(3))\)

First, find the value of \(f(3)\), which is 4. Then evaluate \(g(4)\). Given \(g(4)=-1\), we have: \(g(f(3)) = g(4) = -1\).

Key concepts

Function Evaluation

In mathematics, evaluating a function is a fundamental process that involves substituting a value into the function in place of the variable. For instance, if you have a function \(f(x)\), and you want to evaluate it at \(x = 2\), you would calculate \(f(2)\). This means substituting \(2\) for \(x\) in the function's formula and then performing the arithmetic to find the result. In our exercise, we were given several values for different functions such as \(f(2) = 3\), and we used these to evaluate more complex expressions.

Composing Functions

Composing functions involves creating a new function by combining two or more functions. This means using the output of one function as the input of another. The notation for composing functions \(f\) and \(g\) is \((f \, \b{circ} \, g)(x)\), and it represents \(f(g(x))\). In the provided exercise, we computed several composite functions step by step. For example, to find \(f(g(3))\), we first evaluated \(g(3)\) to get 2, and then used this result to evaluate \(f(2)\), which we knew to be 3. Each step built on the last, demonstrating the principle of composing functions.

Function Values

Understanding function values is crucial for solving problems involving both function evaluation and composition. Function values tell us what output a specific input gives when passed through the function. In the exercise, we were given several function values like \(f(3) = 4\) and \(g(2) = 5\). Using these values, we could determine the results of the more complex expressions by systematically plugging in and reusing known values. This step-by-step use of given function values ensured accurate calculations for composite functions.

Precalculus

Precalculus serves as the foundation for calculus and involves a deep understanding of functions, their properties, and operations on functions. Evaluating and composing functions, as shown in the exercise, are essential skills in precalculus. Mastering these concepts allows students to tackle more complex mathematical problems. Precalculus ties together various areas of algebra and trigonometry, laying the groundwork for the study of limits, derivatives, and integrals in calculus. Therefore, practicing problems involving function evaluation and composition, like the one provided, builds a strong base for further mathematical exploration.