MATH107 Basic Mathematics Quiz
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Questions and Answers

What is the result of the function $F(G(p))$ based on the provided calculations?

  • $4p^2 + 6p + 2$
  • $4p^2 + 8p + 4$
  • $2p^2 + 12p + 2$
  • $4p^2 + 12p + 2$ (correct)

    • What is the inverse function of $f(x) = ax + b$?

  • $ax - b$
  • $ rac{ax}{b}$
  • $ rac{(x-b)}{b}$
  • $ rac{(x-b)}{a}$ (correct)

    • If $f(x) = 3x - 5$ and $g(x) = x^2 - 2x + 1$, what is $p(x)$ defined as?

  • $6x - 5$
  • $x^2 + 1$
  • $3x^2 - 5$
  • $x^2 + x - 4$ (correct)

    • What is the unique solution of the equation $f(x) = 0$ if $f$ is one-to-one?

    <p>$f^{-1}(0)$ (C)</p> Signup and view all the answers

    What is the form of the function $g(h(x))$ if $h(x) = x + 1$ and $g(u) = u^2 - 2u + 10$?

    <p>$x^2 + 9$ (D)</p> Signup and view all the answers

    What is the result of $G(F(1))$ given that $F(1) = 2$?

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

    What is the resultant function $q(x)$ if $h(x) - j(x)$ is calculated with $h(x) = x^3$ and $j(x) = rac{1}{2x^4}$?

    <p>$x^3 - rac{1}{2x^4}$ (D)</p> Signup and view all the answers

    What is the final result of $g(h(2))$, if $g(u) = 2u^3$ and $u = h(x) = x^2 - 2x + 5$?

    <p>$250$ (D)</p> Signup and view all the answers

    What is the domain of the function $y = f(x) = rac{1}{x^2-4}$?

    <p>$ orall x eq -2, 2$ (A)</p> Signup and view all the answers

    Which of the following represents the domain of the function $y = f(x) = ext{sqrt}(x-5)$?

    <p>$[5, ext{inf})$ (D)</p> Signup and view all the answers

    Which operation results in $(f-g)(x)$ if $f(x) = 3x - 1$ and $g(x) = x^2 + 3x$?

    <p>$-1 - x^2$ (D)</p> Signup and view all the answers

    What is the result of $F(G(p))$ if $F(p) = p^2 + 4p - 3$ and $G(p) = 2p + 1$?

    <p>$(2p + 1)^2 + 4(2p + 1) - 3$ (C)</p> Signup and view all the answers

    What is the output of the constant function $h(x) = 2$ for any input?

    <p>$2$ (B)</p> Signup and view all the answers

    Which of the following combinations represents $(fg)(x)$ given $f(x) = 3x - 1$ and $g(x) = x^2 + 3x$?

    <p>$3x^3 + 8x^2 - 3x$ (B)</p> Signup and view all the answers

    What is the result of $(f+g)(x)$ if $f(x) = 3x - 1$ and $g(x) = x^2 + 3x$?

    <p>$x^2 + 6x - 1$ (B)</p> Signup and view all the answers

    What is the definition of a function?

    <p>A mathematical rule that assigns to each input value one and only one output value. (C)</p> Signup and view all the answers

    Which of the following correctly identifies the domain of the function $g(x) = 3x^2 - x + 5$?

    <p>All real numbers. (B)</p> Signup and view all the answers

    How are two functions f and g determined to be equal?

    <p>Both functions must have identical domains and equal outputs for all inputs. (B)</p> Signup and view all the answers

    What is the range of a function?

    <p>The set of all possible output values. (C)</p> Signup and view all the answers

    What will be the output value when evaluating the function $k(x) = \begin{cases} x + 2\text{ if } x\neq1 \ 3\text{ if } x = 1 \ \end{cases}$ at x = 1?

    <p>3 (B)</p> Signup and view all the answers

    What is the value of $g(z)$ if $g(x) = 3x^2 - x + 5$?

    <p>3z^2 - z + 5 (D)</p> Signup and view all the answers

    Which of the following statements about the function $h(x)$ is true?

    <p>$h(x)$ has an output of 0 when x = 1. (C)</p> Signup and view all the answers

    For the function $t = u(v) = 2v^2 - 5v$, what is the value of $u(-5)$?

    <p>75 (C)</p> Signup and view all the answers

    Flashcards

    Function

    A mathematical rule that assigns a unique output value to each input value.

    Domain

    The set of all possible input values for a function.

    Range

    The set of all possible output values for a function.

    Equality of Functions

    Two functions are equal if they have the same domain and produce the same output for every input in their domain.

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    Finding Function Values

    To find the value of a function at a specific input, simply substitute the input value into the function's equation and simplify.

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    Input and Output of a Function

    A function's input, symbolized by the variable 'x' in most cases, determines the corresponding output, symbolized by the variable 'y' or 'f(x)' usually.

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    Function Diagram

    A function is represented by a diagram where arrows connect input values to their corresponding output values.

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    Function Representation

    A function is represented by a rule or equation that expresses the relationship between input and output.

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    Constant Function

    A function where the output value is always the same, regardless of the input.

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    Function Definition

    A function where the output is determined by a specific rule or equation.

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    Rational Function

    A function that can be expressed as a ratio of two polynomials.

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    Absolute Value Function

    A function that measures the distance from zero on a number line, always returning a non-negative value.

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    Operations on Functions (addition, subtraction, multiplication, division)

    Adding, subtracting, multiplying, or dividing functions to create new functions.

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    Composition of Functions

    Applying one function to the output of another function.

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    Domain of a Function

    The input value of a function.

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    Range of a Function

    The set of all possible output values of a function.

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    Inverse Function

    A function that reverses the effect of another function, meaning if you apply one function and then its inverse, you get back to the original input.

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    One-to-One Function

    A function is one-to-one if each output value corresponds to only one input value. In other words, no two different inputs produce the same output.

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    Function Composition

    The process of combining multiple functions to create a new function by applying one function to the output of another.

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    Composite Function Notation

    The notation 'f(g(x))' means that we apply function g to 'x', then apply function f to the output of g.

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    Order of Composition

    When composing functions, the order can matter drastically. The function applied first acts on the input value, and the resulting output is then used as the input for the next function.

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    Finding the Inverse of a Function

    To find the inverse of a function, solve the equation y = f(x) for x in terms of y. The resulting expression for x in terms of y will be the inverse function, denoted as f⁻¹(x).

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    Inverse Function Property

    The result of applying a function 'f' to its inverse 'f⁻¹' (or vice versa) always returns the original input value 'x'.

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    Solving Equations with Inverses

    If a function 'f' is one-to-one, then the equation f(x) = 0 has a unique solution given by x = f⁻¹(0).

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    Study Notes

    Basic Mathematics Course Notes

    • This course is MATH107, Basic Mathematics.
    • Previous week's topics included Introductory Mathematical Analysis, Chapter 3 (Lines, Parabolas, and Systems)
    • Chapter 2 (Functions and Graphs) was covered.
    • Functions assign each input to only one output.
    • The set of all input values is the domain.
    • The set of all output values is the range.
    • Two functions f and g are equal (f = g) if their domains are equal and f(x) = g(x) for all x in the domain.

    Function Equality Examples

    • Example 1 demonstrates determining which functions are equal.
    • Functions f(x), g(x), h(x), and k(x) are examined.
    • The example identifies equal functions based on their relationship and the domain.

    Finding Domain and Function Values

    • Example 3 shows finding the domain and function values.
    • The domain of the function g(x) = 3x² - x + 5 is all real numbers.
    • g(z) = 3z² - z + 5
    • g(r²) = 3r⁴ - r² + 5
    • g(x + h) = 3(x + h)² - (x + h) + 5

    Mathematical Functions

    • A function is a mathematical rule assigning each input value to exactly one output value.
    • The domain is the set of all possible input values.
    • The range is the set of all possible output values.

    Examples of Equations

    • y = x² - 2x+1; values of x are inputs (the domain), y values are outputs (the range).
    • Values of x are inputs. Values of y are outputs/range.
    • If x=1, then y=0
    • Example equation y = 2x² + 5x + 6 to find f(2)=24

    Domain of Functions

    • Example of a function with specified domain: y = f(x) = x²-2x+3. The domain of f is all real numbers.
    • If the equation is y=x²-2x+3, the given domain is for all x; given x²-4≠0 because √(0) is undefined for x≠ ±2. Then D = R {±2} = R₂.
    • For y = √(x-5), the domain is x ≥ 5.
    • For y = √(25 - x²), the range is [-5,5].
    • Another example: y = √(x + 3)/(x² -1); the denominator cannot equal zero, so x ≠ ±1, x ≥ -3.

    Combining Functions

    • Example of combining functions: if f(x) = 3x - 1 and g(x) = x² + 3x
    • Find composite functions like (f+g)(x), (f-g)(x), (fg)(x), g(f(x)), f(g(x))
    • This includes finding solutions for composite operations.

    Composition of Functions

    • Example: F(p) = p² + 4p - 3, G(p) = 2p + 1, H(p) = p. Find F(G(p)), F(G(H(p))), G(F(1))

    Inverse Functions

    • An inverse function is defined as f(f⁻¹(x))=x = f⁻¹(f(x)).
    • Example of an inverse function: f(x) = (x - 1)². Find f⁻¹(x) for x ≥ 1.
    • Solution: Let y = (x - 1)², x - 1 = √y; x =√y + 1; f⁻¹(x) = √x+ 1.

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    Related Documents

    Lecture 9: Basic Mathematics PDF

    Description

    Test your understanding of basic mathematics concepts covered in MATH107. This quiz focuses on functions and graphs, including their properties and equality. Topics from Chapters 2 and 3 will be explored through examples and problem-solving activities.

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