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?

$f^{-1}(0)$ Signup and view all the answers

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

$x^2 + 9$ Signup and view all the answers

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

$5$ 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 $j(x) = rac{1}{2x^4}$?

$x^3 - rac{1}{2x^4}$ Signup and view all the answers

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

$250$ Signup and view all the answers

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

$orall x eq -2, 2$ Signup and view all the answers

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

$[5, ext{inf})$ Signup and view all the answers

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

$-1 - x^2$ Signup and view all the answers

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

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

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

$2$ Signup and view all the answers

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

$3x^3 + 8x^2 - 3x$ Signup and view all the answers

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

$x^2 + 6x - 1$ Signup and view all the answers

What is the definition of a function?

A mathematical rule that assigns to each input value one and only one output value. Signup and view all the answers

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

All real numbers. Signup and view all the answers

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

Both functions must have identical domains and equal outputs for all inputs. Signup and view all the answers

What is the range of a function?

The set of all possible output values. 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?

3 Signup and view all the answers

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

3z^2 - z + 5 Signup and view all the answers

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

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

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

75 Signup and view all the answers

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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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.