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Chapter 1: Problem 15

For Exercises 11-24, evaluate the indicated expression assuming that $$ f(x)=\sqrt{x}, \quad g(x)=\frac{x+1}{x+2}, \quad h(x)=|x-1| $$ $$ (f \circ h)(-3) $$

Short Answer

Expert verified
The composition of the functions f and h at x = -3 is: \((f \circ h)(-3) = f(h(-3)) = f(4) = 2\).

Step by step solution

01

Find h(-3)

To compute the function h(x), plug in x = -3 into the definition of h(x): \(h(-3) = |-3 - 1|\)
02

Simplify h(-3)

Simplify the expression: \(h(-3) = |-4| = 4\) So, \(h(-3) = 4\).
03

Compute (f 鈭� h)(-3) or f(h(-3))

Now that we have calculated the value of h(-3), we can find the value of the composition of the functions f and h at x = -3. To do this, we will plug in h(-3) = 4 into the function f(x): \(f(h(-3)) = f(4)\)
04

Compute f(4)

To compute the function f(x), plug in x = 4 into the definition of f(x): \(f(4) = \sqrt{4}\)
05

Simplify f(4)

Simplify the expression: \(f(4) = 2\) So, the composition of the functions f and h at x = -3 is: \((f \circ h)(-3) = f(h(-3)) = f(4) = 2\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Piecewise Functions
Piecewise functions are like puzzle pieces that fit together to form a complete function. They are defined by different expressions based on different intervals of the input value. Think of it like a set of rules for different parts of the x-axis.
For example, a piecewise function might be written as two separate expressions, one for when x is less than 0 and another for when x is 0 or more. This helps create flexibility in describing how a function behaves in different scenarios.
By using these smaller fragments, piecewise functions can model more complex situations efficiently.
Exploring the Square Root Function
The square root function, represented as \(f(x) = \sqrt{x}\), turns a number into its square root. It's one of the basic operations in mathematics and works only with non-negative values, as the square root of a negative number isn't a real number.
When you see \(\sqrt{4}\), you get 2, because 2 multiplied by itself is 4. This function is important in various fields like geometry and physics, helping in calculations involving areas and velocities.
The square root function creates a smooth, curving line that starts at the origin and continues rightward.
The Role of Absolute Value Functions
An absolute value function, shown as \(h(x) = |x-1|\), measures the distance of a number from zero on the number line without considering direction.
For example, both -4 and 4 have the same absolute value: 4. This is because distance is always non-negative.
Absolute value functions are useful in understanding magnitudes and distances and show up in real-life applications like error measurement and data analysis. Their graphs typically form a V-shape, centered on the x-axis.
Mastering Function Evaluation and Composition
Function evaluation and composition is all about plugging numbers into functions and combining them. When you evaluate a function, you're determining its output for a particular input, like substituting \(-3\) into \(h(x)\) to get 4.
Function composition, denoted as \((f \circ h)(-3)\), involves finding the result of one function and then using that result in another function. It's like solving a math puzzle where each answer leads you to the next clue.
By understanding these operations, you can unlock more complex mathematical relationships and solve intricate problems with ease.

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Most popular questions from this chapter

The result box following Example 2 could have been made more complete by including explicit information about the domain and range of the functions \(g\) and \(h\). For example, the more complete result box might have looked like the one shown here: Shifting a graph up or down Suppose \(f\) is a function and \(a>0 .\) Define functions \(g\) and \(h\) by \(g(x)=f(x)+a\) and \(h(x)=f(x)-a\) Then \- \(g\) and \(h\) have the same domain as \(f\); \- the range of \(g\) is obtained by adding \(a\) to every number in the range of \(f\); \- the range of \(h\) is obtained by subtracting \(a\) from every number in the range of \(f\); the graph of \(g\) is obtained by shifting the graph of \(f\) up \(a\) units; \- the graph of \(h\) is obtained by shifting the graph of \(f\) down \(a\) units. Construct similar complete result boxes, including explicit information about the domain and range of the functions \(g\) and \(h,\) for each of the other three result boxes in this section that deal with function transformations.

A constant function is a function whose value is the same at every number in its domain. For example, the function \(f\) defined by \(f(x)=4\) for every number \(x\) is a constant function. Give an example of three functions \(f, g,\) and \(h\), none of which is a constant function, such that \(f \circ h=g \circ h\) but \(f\) is not equal to \(g\).

Give an example of a function \(f\) such that the domain of \(f\) and the range of \(f\) both equal the set of integers, but \(f\) is not a one-to-one function.

Suppose \(f\) and \(g\) are functions, each of whose domain consists of four numbers, with \(f\) and \(g\) defined by the tables below: $$ \begin{array}{c|c} {x} & {f}({x}) \\ \hline {1} & 4 \\ 2 & 5 \\ 3 & 2 \\ 4 & 3 \end{array} $$ $$ \begin{array}{c|c} x & g(x) \\ \hline 2 & 3 \\ 3 & 2 \\ 4 & 4 \\ 5 & 1 \end{array} $$ Sketch the graph of \(f^{-1}\).

Assume that \(f\) is the function defined by $$ f(x)=\left\\{\begin{array}{ll} 2 x+9 & \text { if } x<0 \\ 3 x-10 & \text { if } x \geq 0. \end{array}\right. $$ Find two different values of \(x\) such that \(f(x)=4\).
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