Problem 2 - Discrete Mathematics with Applications

Question

Let \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=2 x-1\) and \(g(x)=x^{2}+1 .\) Find: $$(f \circ g)(-1)$$

Step-by-Step Solution

Answer

The short answer is: \((f \circ g)(-1) = 3\).

Step 1: Write down given functions

We are given: \(f(x) = 2x - 1\) \(g(x) = x^2 + 1\)

Step 2: Find the composition

We need to find \(f(g(x))\), which means we will substitute the expression for \(g(x)\) into the function \(f(x)\) where \(x\) appears: \(f(g(x)) = 2(g(x)) - 1\)

Step 3: Substitute the expression for \(g(x)\) and simplify

Now, substitute the expression for \(g(x) = x^2 + 1\) into the simplified form of \(f(g(x))\). \(f(g(x)) = 2(x^2 + 1) - 1\) Simplify the expression: \(f(g(x)) = 2x^2 + 2 - 1\) \(f(g(x)) = 2x^2 + 1\)

Step 4: Evaluate the composed function at x = -1

Now we have found the composed function as \(f(g(x)) = 2x^2 + 1\), we can substitute \(x = -1\): \(f(g(-1)) = 2(-1)^2 + 1\) \(f(g(-1)) = 2(1) + 1\) \(f(g(-1)) = 2 + 1\) \(f(g(-1)) = 3\) The value of \((f \circ g)(-1)\) is 3.

Key Concepts

Composite Functions

Understanding composite functions is akin to learning how to follow a multi-step recipe: each step logically flows from the one before. Imagine you are generating a new function by combining two existing ones, much like mixing ingredients to create a novel dish.

When looking at our exercise, the composite function \(f \circ g)(x)\) is like saying 'evaluate function \(g\) first, then plug its output into function \(f\)'. You're essentially performing two function evaluations in sequence: one after the other. Now, to ensure clarity for students—think of \(g(x)\) as the intermediate result, which is then used as the input for \(f(x)\).

Always remember, the order is essential—reversing the functions would yield a completely different outcome, just as using sugar instead of salt can drastically change a dish. To visualize this, if \(g\) turns your input \(x\) into \(x^2 + 1\), \(f\) then transforms that result further by applying its own rule, here \(2x - 1\), leading to our final expression.

Function Evaluation

Evaluating a function for a specific input is like finding out what's in a wrapped gift—you plug in the value, follow the function's rules, and out comes the answer. In our math exercise, we evaluate the composite function \((f \circ g)(-1)\), which requires us to first substitute -1 into \(g(x)\), and then use the output as the new input for \(f(x)\).

This concept of function evaluation is critical for students to master because it's the bridge between abstract function rules and concrete numerical results. Like reading individual words before understanding a sentence, students must become proficient at this basic operation to progress in mathematics.

Quadratic Functions

The quadratic functions you encounter feel like the math world's parabolas, arching gracefully across your graph paper. They are defined by the general form \(ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). In this particular exercise, \(g(x) = x^2 + 1\) is our quadratic function.

Why study them? They model countless phenomena: the arc of a ball thrown in the air, the profitability of selling items, or even the spread of diseases. Compared to linear functions, quadratics add a layer of complexity, featuring not just slopes but curves, peaks, and troughs that invite a deeper level of analysis and appreciation.