Problem 56 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x) = 2 x^{2}, g(x) = x+1\), and \(h(x) = \frac{1}{x}\) . If what is written is an expression, simplify it. If it is an equation, solve it. $$ f(g(x))=10 $$

Step-by-Step Solution

Answer

Therefore, the solutions to the equation are \( x = \sqrt{5} - 1 \) and \( x = -\sqrt{5} - 1 \)

Step 1: Compose the Functions

To simplify this expression, first plug \( g(x) \) into \( f(x) \). Therefore, replace every \( x \) in \( f(x) = 2x^{2} \) with \( g(x) = x+1 \) to get \( f(g(x)) = 2(x+1)^{2} \)

Step 2: Simplify the Equation

Setting this function equal to 10, the equation to solve for \( x \) becomes \( 2(x+1)^{2} = 10 \)

Step 3: Solve for \( x \)

Begin by dividing both sides of the equation by 2, yielding \( (x + 1)^{2} = 5 \). Next, take the square root of both sides to yield \( x + 1 = \sqrt{5} \) or \( x + 1 = -\sqrt{5} \). Finally, subtract 1 from both sides to get \( x = \sqrt{5} - 1 \) and \( x = -\sqrt{5} - 1 \)

Key Concepts

Function Composition

In the realm of calculus, function composition is a binary operation that takes two functions and produces a new function. This might sound complex, but it's essentially about substituting one function into another. When we talk about composing functions like f(x) and g(x), we use the notation f(g(x)), which means 'apply g to x, then apply f to the result of g(x).

Imagine you're first putting on socks (g(x) = x + 1) and then shoes (f(x) = 2x^2). You wouldn't put the shoes on before the socks, right? The order matters, and that's function composition in a nutshell—a series of steps putting on one function after another.

Solving Quadratic Equations

Quadratic equations are like puzzles. They generally take the form ax^2 + bx + c = 0 where a, b, and c are known values, and x represents what we're trying to find. Think of x as the missing piece of the puzzle. To solve for x, you can rearrange the equation and sometimes factor it, or you could use the Quadratic Formula. But, when the equation is already simplified, like (x + 1)^2 = 5, we can use simpler methods – like the square root method – to find x's values.

Simplifying Expressions

Simplifying expressions is like tidying up your room. You want to make everything as neat and straightforward as possible. In math, we combine like terms, apply distribution laws, and do whatever we can to make the expression easier to read and work with. With something like 2(x + 1)^2, simplification isn't just about making it look pretty; it's about preparing the expression so we can solve it more easily. Think of simplification as the warm-up exercise before the main event, which is, in this case, solving the equation.

Square Root Method

Here's a handy tool for your math toolkit: the square root method. It's a great way to solve for x when your equation involves squaring something. If you have an equation like (x + 1)^2 = 5, you can apply the square root to both sides to cancel out the squaring. You'll get two answers: one positive and one negative (\(x + 1 = \pm \sqrt{5}\)). Remember, it's like finding both ends of a stick; one side is positive, the other negative, but both are part of the same stick. After using the square root method, don't forget to solve for x by isolating it on one side of the equation.