Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 1.25

Chapter 1, Problem 1.25

Defining piecewise functions Write a definition of the function whose graph is given <IMAGE>

Verified step by step guidance

Step 1: Identify the different segments of the graph. Look for changes in the slope, discontinuities, or any other features that indicate a new piece of the function.

Step 2: Determine the mathematical expression for each segment. This could be a linear equation, a quadratic, or another type of function, depending on the shape of the graph.

Step 3: Identify the domain for each piece of the function. This involves determining the x-values over which each segment is defined.

Step 4: Write the piecewise function using the expressions and domains identified in the previous steps. Use the format: $ f(x) = \begin{cases} \text{expression 1,} & \text{if } \text{domain 1} \\ \text{expression 2,} & \text{if } \text{domain 2} \\ \end{cases} $

Step 5: Verify the piecewise function by checking that it matches the graph over the specified domains. Ensure continuity or note any discontinuities as needed.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Piecewise Functions

A piecewise function is defined by multiple sub-functions, each applying to a specific interval of the function's domain. This means that the function can take different forms based on the input value. For example, a piecewise function might be defined as f(x) = x^2 for x < 0 and f(x) = 2x + 1 for x ≥ 0, illustrating how the function behaves differently in different regions.

Domain and Range

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). Understanding the domain and range is crucial for defining piecewise functions, as each piece may have its own specific domain, affecting how the function behaves overall.

Graph Interpretation

Interpreting the graph of a function involves analyzing its shape, slopes, and intercepts to understand its behavior. For piecewise functions, the graph may consist of distinct segments, each corresponding to different rules. Recognizing where these segments start and end is essential for accurately defining the function and understanding its continuity and discontinuities.

Recommended video:

06:15

Graphing The Derivative

Related Practice

Textbook Question

Solving equations Solve the following equations.

5(ˣ³) = 29

Textbook Question

Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

ƒ(x) = |2x + 1|

Textbook Question

Suppose ƒ is an even function with ƒ(2) = 2 and g is an odd function with g(2) = -2. Evaluate ƒ(-2) , ƒ(g(2)), and g(ƒ(-2))

views

Textbook Question

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.

logb (√x) / (³√z)

views

Textbook Question

Solve each equation.

$7^{y-3}=50$

Textbook Question

Solving equations Solve the following equations.

7ˣ = 21