Problem 64 - Precalculus : Building Concepts and Connections

Question

In Exercises \(49-66,\) let \(f(x)=x^{2}+x, g(x)=\sqrt{x},\) and \(h(x)=-3 x\) Evaluate each of the following. $$(h \circ f)(-3)$$

Step-by-Step Solution

Answer

\((h \circ f)(-3) = -18\)

Step 1: Apply Function f(x) on Provided Value

First, substitute \(x = -3\) in the function \(f(x) = x^{2} + x\). Calculate the value of \(f(-3)\) which is \((-3)^{2} + (-3) = 9 - 3\). Hence, \(f(-3) = 6\).

Step 2: Apply Function h(x) on Obtained Result

Next, substitute the obtained result \(f(-3) = 6\) in \(h(x) = -3x\). Calculate the value of \(h(6)\) which is \(-3 \times 6\) and equals to \(-18\).

Step 3: Compose and Finalize Solution

When \(h(x)\) is applied to \(f(-3) = 6\), we get \(h(f(-3)) = h(6)= -18\). Hence, \((h \circ f)(-3) = -18\).

Key Concepts

Function Operations

When dealing with functions in mathematics, we often need to perform operations with them, similar to the operations we perform with numbers. Function operations include addition, subtraction, multiplication, division, and composition. For example, if we have two functions, say f(x) and g(x), their sum is given by (f + g)(x) = f(x) + g(x). The same applies to the other operations, where we combine the outputs or values of the functions based on the operation at hand.

When solving exercises involving function operations, the key is first to understand what operation is required and then apply the functions' definitions carefully. Each operation can profoundly affect the outcome and interpretation of the functions. This consideration is particularly important when it comes to the composition of functions which has its unique approach.

Function Composition

Function composition is an operation where the output of one function becomes the input of another function. In notation, if we have two functions f and g, the composition is written as (g \( \circ \) f)(x), which means 'g of f of x'. To evaluate this, you first calculate f(x) and then use this result as the input for g.

In the exercise provided, we are looking at \((h \( \circ \) f)(-3)\). Here's a detailed method to follow:

• First, evaluate f(-3) to get the intermediate result.
• Next, apply h to this result, treating it as the input for h.
• Finally, the result of h at this intermediate value gives us the answer to \((h \( \circ \) f)(-3)\).

Understanding this process is crucial as function composition is a fundamental concept that often occurs in higher mathematics and applications.

Precalculus

Precalculus is an important part of mathematics education, serving as the bridge between algebra and calculus. It lays the foundation for understanding the behavior and properties of functions, their graphs, and more complex mathematical concepts. In precalculus, students encounter various types of functions - linear, quadratic, polynomial, rational, exponential, and logarithmic - and learn how to manipulate and graph them.

Function operations and composition, like in the provided exercise, are part of the precalculus curriculum. These topics prepare students for the type of thinking and problem-solving required in calculus. For instance, when students learn about composition of functions in precalculus, it opens the door to later topics like the chain rule in calculus, which deals with the derivative of composed functions.

To excel in precalculus and be well-prepared for calculus, it's essential to practice these concepts through exercises like the one discussed here. By carefully analyzing functions and their operations, students develop a stronger mathematical intuition and problem-solving skills.