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?

<p>Every vertical line must intersect the graph at most once. (D)</p> Signup and view all the answers

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

<p>$1092.73 (B)</p> Signup and view all the answers

What does the domain of a function refer to?

<p>The set of all possible input values (D)</p> Signup and view all the answers

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

<p>A graph where every vertical line intersects at most once (C)</p> Signup and view all the answers

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

<p>A (B)</p> Signup and view all the answers

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

<p>Each x-value can correspond to only one y-value. (A)</p> Signup and view all the answers

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

<p>16 (A)</p> Signup and view all the answers

What is f(-2) if f(x) = 3x² + x - 1?

<p>5 (B)</p> Signup and view all the answers

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

<p>It has different formulas for separate intervals of x. (D)</p> 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}?

<p>1 (A)</p> 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?

<p>Isolate x through inverse operations. (C)</p> Signup and view all the answers

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

<p>To establish a consistent relationship between inputs and outputs. (C)</p> Signup and view all the answers

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

<p>7 (A)</p> Signup and view all the answers

Flashcards

Function

A rule that assigns each input value exactly one output value.

Domain

The set of all possible input values (x-values) of a function.

Range

The set of all possible output values (y-values) of a function.

Vertical Line Test

A method to determine if a graph represents a function. If any vertical line intersects the graph more than once, it is not a function.

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Function Representation

Different ways to describe a function, including verbally, algebraically (with a formula), graphically, or numerically (with a table).

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Function Formula

An equation that expresses the output value in terms of the input value.

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Graph of a function

A visual representation of a function, where the input values are along the horizontal axis and the output values are along the vertical axis.

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Numerical Representation

Function represented by a table of input and output values.

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Is y a function of x?

A graph represents a function if each input (x-value) corresponds to exactly one output (y-value).

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Independent Variable

The input value (usually x) in a function.

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Dependent Variable

The output value (usually y) in a function, determined by the input.

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f(x)

Function notation, representing the output of a function for a given input x.

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Piecewise function

A function defined by multiple sub-functions, each applying to a specific part of the input domain.

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Function notation f(x)

A way to indicate the output of a function f for the input x.

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Solving for x in a function

Finding the input value(s) that yield a specific output value for a function.

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Evaluating a function

Finding the value of a function (f( x)) when given a specific input value ( x).

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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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