Problem 10 - Applied Calculus

Question

For the functions \(f\) and \(g\) in Problems \(7-10,\) find (a) \(\quad f(g(1))\) (b) \(g(f(1))\) (c) \(\quad f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=1 / x, g(x)=3 x+4$$

Step-by-Step Solution

Answer

(a) \(\frac{1}{7}\), (b) 7, (c) \(\frac{1}{3x+4}\), (d) \(\frac{3}{x} + 4\), (e) \(3 + \frac{4}{t}\)

Step 1: Find g(1)

First, we need to substitute 1 into the function \( g(x) \). The function is given as \( g(x) = 3x + 4 \). So, \( g(1) = 3(1) + 4 = 7 \).

Step 2: Find f(g(1))

Now substitute \( g(1) = 7 \) into the function \( f(x) \). The function \( f(x) = \frac{1}{x} \), so \( f(g(1)) = f(7) = \frac{1}{7} \).

Step 3: Find f(1)

Next, substitute 1 into the function \( f(x) \). Since \( f(x) = \frac{1}{x} \), we have \( f(1) = \frac{1}{1} = 1 \).

Step 4: Find g(f(1))

Substitute \( f(1) = 1 \) into the function \( g(x) \). Thus, \( g(f(1)) = g(1) = 7 \).

Step 5: Find f(g(x))

To find \( f(g(x)) \), substitute \( g(x) = 3x + 4 \) into \( f(x) \). Then \( f(g(x)) = f(3x + 4) = \frac{1}{3x + 4} \).

Step 6: Find g(f(x))

To find \( g(f(x)) \), substitute \( f(x) = \frac{1}{x} \) into \( g(x) \). We have \( g(f(x)) = g\left(\frac{1}{x}\right) = 3\left(\frac{1}{x}\right) + 4 = \frac{3}{x} + 4 \).

Step 7: Find f(t)g(t)

To find \( f(t)g(t) \), use \( f(t) = \frac{1}{t} \) and \( g(t) = 3t + 4 \). Thus, \( f(t)g(t) = \left(\frac{1}{t}\right)(3t + 4) = 3 + \frac{4}{t} \).

Key Concepts

Function Evaluation

Function evaluation is a fundamental math concept that involves substituting a specific value into a function to determine the corresponding result. Consider functions like machines: you input a number, and the function processes it to provide an output. Understanding function evaluation is key to handling more complex tasks, such as working with composite functions and calculating products of functions.

For instance, let's evaluate the function \( g(x) = 3x + 4 \) when \( x = 1 \). By substituting 1 into the function, we calculate \( g(1) = 3(1) + 4 = 7 \). Similarly, to evaluate \( f(x) = \frac{1}{x} \) for \( x = 1 \), you substitute and simplify to find \( f(1) = \frac{1}{1} = 1 \).

This process is straightforward:

• Identify the function.
• Substitute the given value into the function.
• Simplify to find the output.

Remember, function evaluation sets the stage for more advanced operations like compositions and multiplications.

Composition of Functions

Composition of functions is a more advanced concept in which you apply one function to the result of another function. Essentially, you're chaining two operations together. This is expressed mathematically as \( f(g(x)) \) or \( g(f(x)) \). It's like fitting one function's output into another function's input.

When you perform a composition, follow these steps:

• Evaluate the inner function first.
• Use the output of the inner function as the input for the outer function.
• Simplify to find the final result.

Consider the function composition \( f(g(x)) \) with \( f(x) = \frac{1}{x} \) and \( g(x) = 3x + 4 \). Start by evaluating \( g(x) \), which gives \( g(x) = 3x + 4 \). Then substitute this into \( f(x) \) to obtain \( f(g(x)) = \frac{1}{3x + 4} \).

In another example, using \( g(f(x)) \), you substitute \( f(x) = \frac{1}{x} \) into \( g(x) \), resulting in \( g(f(x)) = \frac{3}{x} + 4 \). Composition requires sequential thinking as it combines the functions in a hierarchy.

Function Multiplication

Function multiplication involves multiplying two functions to create a new function, often referred to as the product of functions. The result is a function that represents the multiplication of their respective outputs for any input within their domains. Function multiplication differs from composition, as you're not "nesting" the functions, but rather multiplying them directly.

The general process is:

• Select two functions, \( f(t) \) and \( g(t) \).
• Multiply their outputs for the same input value.
• Simplify the expression if necessary.

In our example, with \( f(t) = \frac{1}{t} \) and \( g(t) = 3t + 4 \), the multiplication \( f(t)g(t) \) results in \( \left(\frac{1}{t}\right)(3t + 4) = 3 + \frac{4}{t} \).

This operation provides a new perspective on how functions relate to one another, expanding your potential to model and analyze scenarios where multiple influences act simultaneously. Function multiplication fits into calculus and other areas of higher math, supporting various analytical methods.