Functions Representations in Algebra Class

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Questions and Answers

Which representation of a function provides a visual interpretation of the relationship?

Verbal (Words)

Algebraic (Formula)

Numerical (Table)

Graphical (correct)

What does the formula A = f(t) = 1000(1.03)^t represent in terms of a banking scenario?

The cumulative interest earned

The total amount in the account after t years (correct)

The annual percentage increase

The initial deposit amount

Which of the following functions does NOT represent y as a function of x?

x^2 + y^2 = 1 (correct)

y = 5 - 2x

y = x^2 - 4

y = 2x + 3

Which condition must be met for a graph to be classified as a function?

Every vertical line must intersect the graph at most once. Signup and view all the answers

In the bank account example, if t = 3 years, what is the value of A?

$1092.73 Signup and view all the answers

What does the domain of a function refer to?

The set of all possible input values Signup and view all the answers

According to the vertical line test, which of the following graphs indicates that a relation is a function?

A graph where every vertical line intersects at most once Signup and view all the answers

If A = f(t) represents a function, which part of the expression signifies the dependent variable?

A Signup and view all the answers

Which statement correctly describes the relationship between x and y in a function?

Each x-value can correspond to only one y-value. Signup and view all the answers

What is the value of h(3) if h(x) = 5(x - 2)² + 1?

16 Signup and view all the answers

What is f(-2) if f(x) = 3x² + x - 1?

5 Signup and view all the answers

How is f(x) described in terms of its different parts in a piecewise function?

It has different formulas for separate intervals of x. Signup and view all the answers

What is the value of g(8) if g(x) is defined as g(x) = {−x, x < −2; 1, −2≤x}?

1 Signup and view all the answers

For the equation f(x) = -2, which of the following is likely the correct approach to solving for x?

Isolate x through inverse operations. Signup and view all the answers

What is the primary purpose of defining a function in mathematical terms?

To establish a consistent relationship between inputs and outputs. Signup and view all the answers

If h(-2) is to be solved from h(x) = -4x + 1, what is the resulting value?

7 Signup and view all the answers

Study Notes

Representing Functions

Functions describe how one quantity depends on another.
Functions can be represented in words, formulas, graphs, or tables.

Verbal Representation (Words)

Example: Depositing $1000 in a bank account with 3% annual interest. The amount (A) in t years can be described as A = f(t) = 1000(1.03)^t.

Algebraic Representation (Formula)

A = f(t) = 1000(1.03)^t, where A is the amount after t years.

Graphical Representation

Visual representation of the function's relationship.
Shows how the output value changes with the input value.
Example: A graph showing the amount of money in the account (A) versus the number of years (t).

Numerical Representation (Table)

Table of input (independent variable) and output (dependent variable) values.
Example: A table showing the amount of money (A) in the account for different years (t).

Function Definition

A function assigns each input value in the domain to exactly one output value in the range.

Example 1: Determining if a formula describes y as a function of x.

Assess if each input (x) value produces exactly one output (y) value.
Example: x² + y² = 1 does not represent y as a function of x.

Vertical Line Test for Functions

Every vertical line intersects a graph at most once to represent a function.

Dependent and Independent Variables

Independent variable (x): Input value.
Dependent variable (y): Output value, determined by the input.

Function Notation

y = f(x) means y is a function of x.
f(x) is the output value when the input is x.

Example 2: Evaluating Functions

Given a function and an input, compute the output value.
Example: If h(x) = 5(x − 2)² + 1, find h(3).

Example 3: Using Functions

Evaluate functions, including given graphs, tables or formulas.
Example: Find f(-2), g(-2), h(-2), solve f(x) = −2 for x.

Piecewise Defined Functions

A function defined by multiple formulas, each applicable over a particular part of the domain.
Example: f(x)=
```
3-x, if x<-1
12x, if x≧-1
```
Evaluate using the formula appropriate to the input value's position in the domain.

Domain and Range of Functions

Domain: Set of all possible input values (x-values).
Range: Set of all possible output values (y-values).
Example: Find the domain of f(x)=

x+5
- ----
x-2

Find the range of g(x)=x² + a .

Example 4: Piecewise Functions, evaluation with given input

Example 5: Simplifying function expressions

Simplify composed functions
Example: If f(x)=x²+2x, find (f(x+h) - f(x)) / h

Example 6: Finding domain and range from a graph

Example 7: Finding the domains of different functions.

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Description

This quiz explores different ways to represent functions, including verbal descriptions, algebraic formulas, graphical illustrations, and numerical tables. Understand how these representations illustrate the relationship between quantities. Prepare to test your knowledge on function representations used in algebra!