Finding Domain of Functions Quiz

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Questions and Answers

What is the domain of the function defined by the equation y = 2x + 1 / (x - 1)?

{x: x ∈ R, x ≠ -1}

{x: x ∈ R}

{x: x ∈ R, x ≠ 0}

{x: x ∈ R, x ≠ 1} (correct)

Which of these functions has a domain that includes all real numbers?

y = (2x + 1)/(3 + x)

y = 2x + 1 / (x - 1)

y = 2x - y = 5

y = |x| + 1 (correct)

For the equation y = 4 - x², what is the domain of the function?

{x: x ∈ R, x ≠ 2}

{x: x ∈ R, x ≠ 0}

{x: x ∈ R} (correct)

{x: x ∈ R, x ≠ -2}

What restrictions on x are introduced by the equation 3y + xy = 2x + 1?

x ≠ -3 Signup and view all the answers

When rewriting the equation x² + y - 4 = 0 to find y, which of the following is true?

y can equal any value as x varies across all real numbers. Signup and view all the answers

What is the domain of the function y = 2x + 1?

{x: x ∈ R} Signup and view all the answers

What is the domain of the function y = √(x + 1)?

{x: x ≥ -1} Signup and view all the answers

For the relation x² + y² = 1, what is the domain in terms of x?

{x: -1 ≤ x ≤ 1} Signup and view all the answers

What is the domain of the quadratic function y = x² - 2x + 2?

{x: x ∈ R} Signup and view all the answers

Which of the following statements is true regarding the function y = x² - 2x + 2?

The domain of the function is all real numbers. Signup and view all the answers

What is the domain of the function defined by the equation $y = 2x + 1$?

{x | x ∈ R} Signup and view all the answers

For the function $y = x^2 - 2x + 2$, what is the domain?

{x | x ∈ R} Signup and view all the answers

What is the domain of the relation defined by the equation $x^2 + y^2 = 1$?

{x | -1 ≤ x ≤ 1} Signup and view all the answers

What is the domain for the function $y = rac{1}{x-1}$?

{x | x ≠ 1} Signup and view all the answers

For the function $y = ext{√}(x + 1)$, what is the domain?

{x | x ≥ -1} Signup and view all the answers

Which of the following sets represents a function?

{(1,2), (2,2), (3,5), (4,5)} Signup and view all the answers

Which statement correctly describes the vertical line test?

A graph that passes the vertical line test can be classified as a function. Signup and view all the answers

Which of the following graphs would not be considered a function according to the vertical line test?

A circle centered at the origin. Signup and view all the answers

What would be the outcome if a vertical line intersects a graph at two different points?

The graph fails the function test. Signup and view all the answers

Which of the following examples represents a valid function relationship?

{(1,1), (2,2), (3,3), (4,5)} Signup and view all the answers

What is the result of evaluating f(3x - 1) if f(x) = 2x + 1?

6x - 1 Signup and view all the answers

What is the value of g(1.5) if g(x) = x^2 - 2x + 2?

1.75 Signup and view all the answers

What is the final value of g(1.5) if g(x) = √x + 1?

1.58 Signup and view all the answers

What is the output of r(1.5) for the function r(x) = (2x + 1)/(x - 1)?

8 Signup and view all the answers

How is the value of f(1.5) calculated for the function f(x) = |x| + 1?

2 Signup and view all the answers

What is the sum of the functions f(x) and g(x)?

7x - 1 Signup and view all the answers

What is the difference (f - g)(x) given f(x) = 3x - 5 and g(x) = 4x + 6?

-x - 11 Signup and view all the answers

What is the expression for (f + h)(x) when f(x) = 2x² + 1 and h(x) = 7x - 11?

2x² + 7x - 10 Signup and view all the answers

What defines a function in relation to input and output values?

Each input value must be associated with one and only one output value. Signup and view all the answers

What is the result of (f - h)(x) for f(x) = 2x² + 1 and h(x) = 7x - 11?

2x² - 7x + 12 Signup and view all the answers

Which of the following correctly describes the 'domain' in a relation?

The set of all x or input values. Signup and view all the answers

Which of the following represents the operation of (f + g)(x)?

7x - 1 Signup and view all the answers

In the given mapping diagram, what does the 'range' represent?

The set of all y or output values corresponding to the inputs. Signup and view all the answers

Based on the examples provided, which scenario does NOT represent a function?

Input of 4 mapping to outputs 30 and 40. Signup and view all the answers

How can you determine if a relation is classified as a function?

If each input is linked to exactly one output. Signup and view all the answers

What is the result of the expression (v + g)(x)?

2x² + 7x - 4 Signup and view all the answers

Which of the following expressions correctly represents (p - f)(x)?

x - 10 Signup and view all the answers

What is the output of the function (f * p)(x)?

2x² - x - 21 Signup and view all the answers

What does the expression (f - p)(x) equal?

-x + 10 Signup and view all the answers

What is the correct result for the expression v / g(x) provided g(x) ≠ 0?

Incomplete for this problem Signup and view all the answers

What is the value of f(-2) for the piecewise function given?

-3 Signup and view all the answers

Which expression represents f(0) in the piecewise function provided?

6 Signup and view all the answers

Which of the following values is not possible for the functions g(x) or r(x)?

1 Signup and view all the answers

What is the result of f(3) based on the provided piecewise function?

31 Signup and view all the answers

What is the value of f(2) in the piecewise function?

12 Signup and view all the answers

Study Notes

Finding the Domain of a Function

For the function ( y = \frac{2x+1}{x-1} ), the domain excludes ( x = 1 ) since the denominator cannot be zero. Domain: {x | x ∈ R, x ≠ 1}
The function ( y = |x| + 1 ) has a domain of all real numbers: {x | x ∈ R}
For the linear equation ( 2x - y = 5 ), it can be rewritten as ( y = 2x - 5 ) with a domain of {x | x ∈ R}
The quadratic function ( y = x^2 - 9 ) also has a domain of all real numbers: {x | x ∈ R}
For ( x^2 + y - 4 = 0 ), solved as ( y = 4 - x^2 ), the domain remains all real numbers: {x | x ∈ R}
For ( 3y + xy = 2x + 1 ), rearranging gives ( y = \frac{2x + 1}{3 + x} ). The domain excludes ( x = -3 ): {x | x ∈ R, x ≠ -3}

Understanding Relations & Functions

A relation is a rule connecting inputs (domain) with outputs (range).
The domain consists of all possible input values, while the range includes all possible output values.
An ordered pair represents a combination of an input and its corresponding output.
A set is a collection of distinct objects sharing a characteristic.
A function is a specific type of relation where each input corresponds to exactly one output.

Graphical Representation

The Vertical Line Test determines if a relation is a function; if a vertical line intersects the graph in more than one point, it is not a function.
Graph examples show intersections: circles are not functions, but lines are.

Evaluation of Functions

Evaluating a function means substituting a value from its domain into the function.
Given ( f(x) = 2x + 1 ) and evaluating at ( x = 1.5 ) yields ( f(1.5) = 4 ).
For ( g(x) = x^2 - 2x + 2 ), evaluating at ( x = 1.5 ) gives ( g(1.5) = 1.75 ).
Evaluating piecewise functions requires using the appropriate case based on the input value.

Operations on Functions

Addition/Subtraction:
- For functions ( f(x) = 3x - 5 ) and ( g(x) = 4x + 6), results in ( (f+g)(x) = 7x - 1 ).
Multiplication/Division:
- ( (f * g)(x) = f(x) * g(x) ) and ( (f / g)(x) = f(x) / g(x) ) where ( g(x) ≠ 0 ).

Additional Examples of Piecewise Functions

For ( f(x) = { 3x - 8 \text{ if } x ≥ 2; 4x + 5 \text{ if } x < 2 } ):
- Evaluating at different inputs (e.g. ( f(-2) ), ( f(2) ), ( f(5) )) leads to specific outputs.
Evaluating piecewise functions requires careful consideration of the defined intervals to find outputs.

Rules for Evaluating Domains

A function's domain is limited by restrictions, such as avoiding negatives under square roots and ensuring denominators are not zero.

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Description

Test your understanding of identifying the domain of various functions through this quiz. Evaluate functions given in different forms, including rational, absolute value, linear, and quadratic equations. Determine which values are excluded from the domain.