Problem 53 - Precalculus with Limits

Question

In Exercises 53-60, find two functions \(f\) and \(g\) such that \((f \circ g)(x) = h(x)\). (There are many correct answers.) \(h(x) = (2x + 1)^2\)

Step-by-Step Solution

Answer

There are many correct combinations of functions \(f\) and \(g\). One possible combination is \(f(x) = x^2\) and \(g(x) = 2x + 1\), because \((f \circ g)(x) = f(g(x)) = (2x + 1)^2 = h(x)\).

Step 1: Identify the Inner Function

Analyzing the function \(h(x) = (2x + 1)^2\), it is observed that the expression within the square, \(2x + 1\), could be considered as an inner function. So, let's denote it as \(g(x) = 2x + 1\).

Step 2: Identify the Outer Function

Then the outer function is a squaring function, because \(h(x)\) equally squares whatever \(g(x)\) produces. Therefore, let's denote it as \(f(x) = x^2\).

Step 3: Verify the Composition of Functions

For the final step, verify that the composition of \(f\) and \(g\) produces \(h(x)\). This means \((f \circ g)(x) = f(g(x)) = h(x)\). In effect, compute \(f(g(x)) = f(2x + 1) = (2x + 1)^2\), which is indeed equal to \(h(x)\). Hence, the chosen functions \(f\) and \(g\) are correct.

Key Concepts

Function Composition

The concept of function composition is like a 'function of a function', where the output of one function becomes the input of another. Imagine a scenario where you have two separate processes: the first process modifies an object, and the second process takes that modified object and changes it further. In mathematics, function composition works exactly the same way.

When we have two functions, say, f and g, and we want to combine them, we use the notation (f \(\circ\) g)(x), which is read as 'f composed with g of x'. To execute this composition, we first evaluate g(x), and then we take that result and plug it into f. So essentially, (f \(\circ\) g)(x) is the same as f(g(x)). This allows us to create new functions from existing ones and can lead to increasingly complex functions that stem from simpler components.

Inner and Outer Functions

In the realm of composition, understanding inner and outer functions is key to unraveling how complex functions are built. The 'inner' function is the one that is applied first, and its range (output) becomes the domain (input) for the 'outer' function.

For example, if we have a composition (f \(\circ\) g)(x), g is the inner function, while f is the outer function. To visualize this, consider g as the innermost part of a nesting doll, and f as the outer layer that encases it. You first open the outer doll (f) to see what's inside (g(x)). The process is sequential, and order matters; reversing the functions would result in a different output, if it's defined at all.

Quadratic Functions

Among the family of algebraic functions, quadratic functions take a special place with their characteristic parabolic graphs. They are defined by a standard form y = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is called a parabola, which opens upwards if a > 0 and downwards if a < 0.

Quadratic functions model various real-world phenomena, such as projectile motion, the shape of satellite dishes, or options pricing in finance. The function (2x + 1)^2 from the exercise is a specific type of quadratic function where the variable x is first transformed by g(x) before being squared by the outer function f(x).

Algebraic Functions

Algebraic functions form a broad category that includes any function which can be constructed using algebraic operations, such as addition, subtraction, multiplication, division, and taking roots among real numbers. These operations can involve whole numbers, fractions, and irrational numbers.

When dissecting algebraic functions, we can often decompose them into simpler parts or 'building blocks'. By understanding how to construct and deconstruct algebraic expressions, students gain valuable insights into problem-solving and function analysis. The exercise we're discussing exemplifies how algebraic operations can create a quadratic function, which we then analyze through the lens of composition.