Functions and Relations Quiz

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Questions and Answers

Which of the following correctly defines the domain of a relation?

The set of first elements of the ordered pairs. (correct)

The set of all second elements of ordered pairs.

The set of values that make the second element true.

The set of ordered pairs where the first element is unique.

What constitutes a function according to the definition provided?

A relation with at least one ordered pair having the same first and second elements.

A relation where no two ordered pairs have the same first coordinates. (correct)

Any relation where all ordered pairs are unique.

A relation where two ordered pairs can share the same second element.

When sketching the graph of a relation with inequalities, what is the first step?

Identify the range based on the inequalities.

Determine the intersection of the regions.

Plot only the points where the inequalities are true.

Sketch the regions of each inequality in the same coordinate system. (correct)

In the provided relation A: R = {(x, y): y ≥ x and y ≥ -2x + 4}, what will the region look like?

An area between the lines defined by the inequalities. Signup and view all the answers

Which of the following is true about the range of a function?

It is the set of second elements of the ordered pairs. Signup and view all the answers

If a relation has ordered pairs with the same first element but different second elements, what is it classified as?

A general relation. Signup and view all the answers

In the inequality relation B: R = {(x, y): x - 2y ≤ 0 and |x| ≥ 1}, which part is true?

x values must exceed 1 in magnitude. Signup and view all the answers

What condition must be satisfied for a relation to be considered a function?

No first elements repeat in the ordered pairs. Signup and view all the answers

What is the y-intercept of the function defined by the equation f(x) = 3x^2 + 2?

(0, 2) Signup and view all the answers

In the quadratic function f(x) = ax^2 + bx + c, what can be stated about the coefficient 'a'?

'a' represents the leading coefficient and must not be equal to 0. Signup and view all the answers

When is the x-intercept of a function f(x) found?

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

Which of the following forms represents a quadratic function?

f(x) = 4x^2 - 7 Signup and view all the answers

From the function f(x) = (x - 3)(x + 3), how would this be simplified in standard form?

f(x) = x^2 - 9 Signup and view all the answers

What is the range of values for x if you are evaluating the function in the interval [-2, 2]?

-2 ≤ x ≤ 2 Signup and view all the answers

What shape does the graph of the quadratic function f(x) = -2x^2 take?

A concave down parabola Signup and view all the answers

Which of the following points represents the x-intercept of the function graphed by f(x) = 2x^2 + 4x + 2?

(-2, 0) Signup and view all the answers

Which of the following pairs correctly represents a function?

R = {(3, 5), (4, 5), (5, 6)} Signup and view all the answers

What is the domain of the function R1 if given as R1 = {(1, a), (3, b), (5, a)}?

{1, 3, 5} Signup and view all the answers

Which of the following best describes the range of the function R3 = {(1, a), (3, b), (5, c)}?

{a, b, c} Signup and view all the answers

What does f(x) represent in function notation?

The output corresponding to the input x. Signup and view all the answers

If a relation R is defined as R = {(x, y): y is the mother of x}, what is true about this relation?

It is a function since each child has one mother. Signup and view all the answers

Which statement is true about the function notation f(x) = 2x + 1?

f(3) = 7 Signup and view all the answers

If a relation pairs 7 with both 8 and 6, what can be concluded about it?

It is not a function since it has multiple outputs for the same input. Signup and view all the answers

In the function f(x) = x - 3, what is the functional value at x = 5?

2 Signup and view all the answers

What are the values of the function when x is -1 for the function f(x) = 2x^2 + 3?

11 Signup and view all the answers

What is the range of the function f(x) = 2x^2 + 3?

{y: y ≥ 3} Signup and view all the answers

What is the vertex of the function f(x) = -2x^2 + 3?

(0, 3) Signup and view all the answers

Which statement is true regarding the graph of the function when a > 0?

The graph opens upward. Signup and view all the answers

What happens to the axis of symmetry for the function f(x) = 2x^2 + c as c increases?

It remains constant. Signup and view all the answers

For the function f(x) = 2x^2 - 3, what is the value of f(2)?

5 Signup and view all the answers

Which graph characteristic is true for the function f(x) = -2x^2 - 3?

The range is y ≤ -3. Signup and view all the answers

What is the effect of changing the coefficient a from positive to negative in the function f(x) = ax^2 + c?

It alters the direction in which the parabola opens. Signup and view all the answers

What is the highest or lowest point of a parabola called?

Vertex Signup and view all the answers

What does the axis of symmetry of a parabola represent?

A vertical line through the vertex Signup and view all the answers

If a quadratic function is defined by the equation $f(x) = a(x - h)^2 + k$ and $a < 0$, what can be concluded about the graph?

The vertex is located at (h, k). Signup and view all the answers

What happens to the graph of a quadratic function when the absolute value of $a$ increases?

The graph becomes narrower. Signup and view all the answers

When the graph of a quadratic function opens upward, what type of value does the vertex represent?

Minimum value Signup and view all the answers

In the equation $f(x) = a(x - h)^2 + k$, how do you determine the direction the parabola opens?

By the sign of $a$ Signup and view all the answers

If a quadratic function is given by $f(x) = 2x^2 + 3$, what is the direction of its graph?

Opens upward Signup and view all the answers

Shifting the graph of $f(x) = ax^2$ horizontally to the right is achieved by which transformation?

Using $(x - h)$ Signup and view all the answers

Study Notes

Relations

A relation is a set of ordered pairs.
The domain of a relation is the set of all first elements of the ordered pairs.
The range of a relation is the set of all second elements of the ordered pairs.
A relation is a function if and only if no two ordered pairs have the same first element.

Functions

A function is a relation in which no two ordered pairs have the same first element.
The domain of a function is the set of all possible input values.
The range of a function is the set of all possible output values.
The functional value of f at x is denoted by f(x) and is called the image of x under the function f.

Graphing Relations

To sketch the graph of a relation with two or more inequalities, sketch the regions of each inequality on the same coordinate system.
The intersection of the regions is the graph of the relation.

Graphs of Linear Functions

The graph of a linear function is a straight line.
The slope-intercept form of a linear function is y = mx + b, where m is the slope and b is the y-intercept.
The y-intercept is the point where the graph crosses the y-axis.
The x-intercept is the point where the graph crosses the x-axis.

Graphs of Quadratic Functions

A quadratic function is a function defined by f(x) = ax^2 + bx + c, where a, b, and c are real numbers and a ≠ 0.
The graph of a quadratic function is a parabola.
The vertex is the lowest or highest point of the parabola.
The axis of symmetry is the vertical line that passes through the vertex.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
The graph of f(x) = a(x - h)^2 + k can be obtained by shifting the graph of f(x) = ax^2, h units to the right if h > 0, and h units to the left if h < 0, and k units up if k > 0, and k units down if k < 0.

Minimum and Maximum Values of Quadratic Functions

If the graph of a quadratic function opens upward, the function has a minimum value.
If the graph of a quadratic function opens downward, the function has a maximum value.
The minimum or maximum value of a quadratic function is obtained at the vertex of its graph.

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Description

Test your understanding of relations and functions, including their definitions, domains, ranges, and the characteristics that distinguish a function from a general relation. Additionally, explore how to graph these relations and linear functions effectively.