Piecewise and Absolute Value Functions

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Questions and Answers

What is the domain of the function $f(x) = 2 - 0.4x$?

Only integer values

All negative numbers

All real numbers (correct)

All positive numbers

If $f(t) = 2t - t^2$, what does the graph of the function represent over time for a pie cooling?

A parabolic curve that rises then falls (correct)

Consistent increase in temperature

Straight line with no change in temperature

Consistent decrease in temperature

How does the height of grass change over a four-week period as modeled by the homeowner's mowing schedule?

Exponential growth without interruption

Fluctuating growth depending on mowing frequency (correct)

Constant decrease in height

Linear growth of height with no reductions

What does the function $H(t) = 3x - x^2$ suggest about height over time?

Height reaches a maximum then decreases Signup and view all the answers

If an airplane travels 400 miles in one hour, what is its average speed?

400 miles per hour Signup and view all the answers

What effect does time $t$ have on the function $f(t) = 4 - t^2$ in relation to temperature changes?

Temperature decreases with increasing time Signup and view all the answers

What is the output of the function f for the input x = 0?

1 Signup and view all the answers

How does the function $G(x) = 3x - x^2$ behave as $x$ increases?

It reaches a peak then falls Signup and view all the answers

What is the value of f when x = 1?

0 Signup and view all the answers

For x > 1, the function f(x) is defined as which of the following expressions?

x^2 Signup and view all the answers

Given the function $t(x) = x^2 + 1$ for $x eq 0$, how does this function behave for negative $x$ values?

It produces only positive values Signup and view all the answers

What does the solid dot on the graph of f at the point (1,0) indicate?

The point is included in the function. Signup and view all the answers

What is the slope of the line representing f(x) for x < 1?

-1 Signup and view all the answers

Which statement is true about the value of f for any x such that |x| > 0?

f(x) is always a positive number. Signup and view all the answers

What is the behavior of f(x) when x approaches 1 from the left?

f(x) approaches 0. Signup and view all the answers

The absolute value function a is defined as which of the following?

a = a for all numbers a. Signup and view all the answers

What is the y-intercept of the linear function T?

20 Signup and view all the answers

What value of T corresponds to h = 1?

10 Signup and view all the answers

What is the slope (m) of the linear function T?

10 Signup and view all the answers

How can the linear function T be expressed in terms of h?

T = -10h + 20 Signup and view all the answers

If h increases by 1, what is the change in value of T?

-10 Signup and view all the answers

What characterizes an even function?

It satisfies the condition f(-x) = f(x). Signup and view all the answers

Which of the following examples is an even function?

f(x) = x^2 Signup and view all the answers

What does the graph of an even function exhibit regarding its symmetry?

Symmetric with respect to the y-axis. Signup and view all the answers

In the function f(x) = -x^2, what can be concluded about its evenness?

It is an even function. Signup and view all the answers

Why are functions like step functions described as jumping?

They change values with no continuity. Signup and view all the answers

Which of the following properties does NOT define an even function?

f(x) = f(x + 2) for all x. Signup and view all the answers

What does the expression f(-x) indicate in relation to function analysis?

The value of the function at its negative input. Signup and view all the answers

How can the symmetry of an even function be visually confirmed?

By seeing if the graph mirrors across the y-axis. Signup and view all the answers

What type of graph would the line segment joining the points (-1, -3) and (-5, 7) produce?

A straight line Signup and view all the answers

If an even function includes the point (-5, 10), which of the following points must also be included?

(5, 10) Signup and view all the answers

For the even function graph, if the point (-5, 3) is included, what must be the corresponding point?

(5, 3) Signup and view all the answers

What shape does the bottom half of the parabola given by the equation x = - (y - 1)^2 + 0 represent?

A downward-facing parabola Signup and view all the answers

How is the perimeter of a rectangle with a fixed area of 16 m² expressed in terms of its length?

P = 2L + 16/L Signup and view all the answers

If the area of the rectangle is fixed at 16 m², what expression provides the perimeter in relation to the sides?

P = 2(W + L) Signup and view all the answers

Which of these best describes how to find the expression for the top half of the circle defined by the equation x^2 = - (y - 2)^2 + 4?

Rearrange to solve for y in terms of x Signup and view all the answers

What is the formula for the area of the rectangle with a fixed perimeter of 20 m expressed in terms of one side's length?

A = L(10 - L) Signup and view all the answers

Study Notes

Function Domain

The domain of a function defines all possible input values for which the function can produce a valid output.
In the case of the function $f(x) = 2 - 0.4x$, the domain encompasses all real numbers because there are no restrictions on the input value $x$.

Function Graphs and Time

The graph of a function can represent how a quantity changes over time.
For example, the function $f(t) = 2t - t^2$ models the cooling of a pie, with $t$ representing time.
The graph would depict the temperature change with time as the pie cools.

Function Behavior and Growth

Functions can be used to model real-world scenarios like the growth of grass.
The function $G(x) = 3x - x^2$ would describe the height of the grass over time, with $x$ representing the number of weeks.
As $x$ increases, the function's behavior would show the grass growing, reaching a peak height, and then eventually decreasing.

Function Output and Input

For a given function, the output is the value the function produces for a specific input value.
For the function $f(x) = 2 - 0.4x$, when the input $x = 0$, the output $f(0) = 2$.

Function Values and Expressions

The value of a function at a specific input is determined by substituting the value into the function's expression.
For example, if $f(x) = 2 - 0.4x$, then $f(1) = 1.6$.

Function Behavior and Graphs

The graph of a function can provide insights into its behavior.
For example, the graph of the function $f(x)$ might show a change in its slope at a certain point, indicating a shift in its rate of change.

Absolute Value Functions

An absolute value function takes the magnitude of a number, disregarding its sign.
The absolute value function, denoted as $|x|$, gives the distance of $x$ from zero.

Linear Functions and Slope

A linear function is represented by a straight line on a graph.
Its slope (m) defines the steepness of the line, indicating the rate of change.

Function Symmetry

Even functions exhibit symmetry about the y-axis.
This means that for any input $x$, the function's output is the same as its output for $-x$.

Even Functions and Examples

An even function is characterized by the property $f(x) = f(-x)$ for all values of $x$.
For example, the function $f(x) = -x^2$ is an even function as $f(x) = f(-x)$ holds true for all values of $x$.

Function Graphs and Symmetry

The graph of an even function exhibits symmetry about the y-axis.
This means that the graph can be reflected across the y-axis and the resulting image will be identical to the original graph.

Function Analysis and Symmetry

The expression $f(-x)$ is used to analyze the symmetry of a function.
If $f(-x) = f(x)$, the function is even.
If $f(-x) = -f(x)$, the function is odd.

Graphing Line Segments

A line segment connecting two points on a graph is represented by a straight line.
The slope of the line segment can be calculated using the formula (y2 - y1) / (x2 - x1).

Even Functions and Symmetry Properties

Even functions have symmetry about the y-axis. If a point (a, b) exists on the graph of an even function, then the point (-a, b) also exists on the graph.

Parabola Shapes and Equations

The equation x = - (y - 1)^2 + 0 defines a parabola that opens to the left.
The bottom half of the parabola is represented by the part of the curve below the vertex.

Perimeter of Rectangles with Fixed Area

If a rectangle has a fixed area, its perimeter can be expressed as a function of its length.
For example, if the area of a rectangle is 16 m², the perimeter can be expressed as P = 2l + 32/l, where l is the length of the rectangle.

Finding the Top Half of a Circle

The equation x^2 = - (y - 2)^2 + 4 defines a circle centered at (0, 2) with a radius of 2.
To find the expression for the top half of the circle, solve for y.

Area of Rectangles with Fixed Perimeter

If a rectangle has a fixed perimeter of 20 m, its area can be expressed as a function of one side's length.
Let l represent the length of the rectangle, then the area can be expressed as A = 10l - l^2.

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Description

This quiz explores key concepts in piecewise defined functions, absolute value functions, and characteristics of even and odd functions. It includes definitions, examples, and graphical representations of these function types, providing a comprehensive understanding of their properties.