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Questions and Answers
Which equation does NOT represent $y$ as a function of $x$?
Which equation does NOT represent $y$ as a function of $x$?
- $x^2 + y^2 = 4$ (correct)
- $y = |x|$
- $y = \sqrt{x}$
- $y = x^2 + 5$
Given $g(t) = 5t^2 - 2t + 3$, what is the value of $g(-2)$?
Given $g(t) = 5t^2 - 2t + 3$, what is the value of $g(-2)$?
- 19
- 27 (correct)
- 3
- -21
If $V(r) = \frac{4}{3}\pi r^3$, what is $V(3)$?
If $V(r) = \frac{4}{3}\pi r^3$, what is $V(3)$?
- $12\pi$
- $108\pi$
- $36\pi$ (correct)
- $9\pi$
Given $f(x) = \sqrt{x + 5} + 1$, find $f(-4)$.
Given $f(x) = \sqrt{x + 5} + 1$, find $f(-4)$.
Let $q(x) = \frac{1}{x+2}$. What is the value of $q(-2)$?
Let $q(x) = \frac{1}{x+2}$. What is the value of $q(-2)$?
A piecewise function is defined as follows: $f(x) = \begin{cases} x^2, & x < 1 \ 3x - 1, & x \geq 1 \end{cases}$. What is the value of $f(1)$?
A piecewise function is defined as follows: $f(x) = \begin{cases} x^2, & x < 1 \ 3x - 1, & x \geq 1 \end{cases}$. What is the value of $f(1)$?
Given $f(x) = \begin{cases} -2x + 3, & x < -2 \ x^2 - 1, & -2 \leq x < 2 \ 5, & x \geq 2 \end{cases}$, what is the value of $f(-2)$?
Given $f(x) = \begin{cases} -2x + 3, & x < -2 \ x^2 - 1, & -2 \leq x < 2 \ 5, & x \geq 2 \end{cases}$, what is the value of $f(-2)$?
Let $f(x) = x^2 + 3$. Complete the following table:
x
-2
-1
0
1
2
f(x)
?
?
?
?
?
Let $f(x) = x^2 + 3$. Complete the following table:
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| f(x) | ? | ? | ? | ? | ? |
If $h(t) = \frac{t}{t+1}$, complete the following table:
t
-2
-1
0
1
2
h(t)
?
?
?
?
?
If $h(t) = \frac{t}{t+1}$, complete the following table:
| t | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| h(t) | ? | ? | ? | ? | ? |
Find the zero(s) of the function $f(x) = -2x + 10$.
Find the zero(s) of the function $f(x) = -2x + 10$.
Determine the zero(s) of the function $f(x) = \frac{8x - 4}{5}$.
Determine the zero(s) of the function $f(x) = \frac{8x - 4}{5}$.
What are the x-intercepts of the function $f(x) = x^2 - 25$?
What are the x-intercepts of the function $f(x) = x^2 - 25$?
Determine the zeros of the function $f(x) = x^2 + 4x - 12$.
Determine the zeros of the function $f(x) = x^2 + 4x - 12$.
Solve for $x$ when $f(x) = 0$, given $f(x) = x^3 - 4x$.
Solve for $x$ when $f(x) = 0$, given $f(x) = x^3 - 4x$.
Find the zeros of the function $f(x) = x^3 + 2x^2 - x - 2$.
Find the zeros of the function $f(x) = x^3 + 2x^2 - x - 2$.
Determine the point(s) where $f(x) = g(x)$, when $f(x) = x^2 - 4x + 4$ and $g(x) = x + 2$.
Determine the point(s) where $f(x) = g(x)$, when $f(x) = x^2 - 4x + 4$ and $g(x) = x + 2$.
Solve for $x$ when $f(x) = g(x)$, given $f(x) = x^4 - 4x^2$ and $g(x) = 0$.
Solve for $x$ when $f(x) = g(x)$, given $f(x) = x^4 - 4x^2$ and $g(x) = 0$.
Find the value(s) of $x$ for which $f(x) = g(x)$, where $f(x) = \sqrt{x}$ and $g(x) = x - 2$.
Find the value(s) of $x$ for which $f(x) = g(x)$, where $f(x) = \sqrt{x}$ and $g(x) = x - 2$.
Determine the domain of the function $f(x) = 7x^3 + 3x^2 - 9x + 11$.
Determine the domain of the function $f(x) = 7x^3 + 3x^2 - 9x + 11$.
Specify the domain of the function $g(y) = \sqrt{y - 3}$.
Specify the domain of the function $g(y) = \sqrt{y - 3}$.
Identify the domain of the function $f(t) = \frac{t}{t - 5}$.
Identify the domain of the function $f(t) = \frac{t}{t - 5}$.
State the domain of the function $h(x) = \frac{5}{x^2 - 4x + 3}$
State the domain of the function $h(x) = \frac{5}{x^2 - 4x + 3}$
What is the domain of the function $f(s) = \sqrt{s - 2}$?
What is the domain of the function $f(s) = \sqrt{s - 2}$?
Find the domain of $f(x) = \frac{x + 2}{\sqrt{x - 4}}$
Find the domain of $f(x) = \frac{x + 2}{\sqrt{x - 4}}$
Crazy Hard: Determine the largest possible domain of the function $f(x) = \frac{\sqrt{9-x^2}}{\log(x+1)}$, expressing your answer in interval notation.
Crazy Hard: Determine the largest possible domain of the function $f(x) = \frac{\sqrt{9-x^2}}{\log(x+1)}$, expressing your answer in interval notation.
Insanely Hard: Find the domain of the function $f(x) = \sqrt{\log_{10}(\frac{5x - x^2}{4})}$
Insanely Hard: Find the domain of the function $f(x) = \sqrt{\log_{10}(\frac{5x - x^2}{4})}$
Flashcards
What is a function?
What is a function?
A relation where each input (x) has only one output (y).
What does evaluate a function mean?
What does evaluate a function mean?
Find the value of the function at a specific input.
What are zeros of a function?
What are zeros of a function?
Values of x for which f(x) equals zero.
What is the domain of a function?
What is the domain of a function?
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What is a piecewise function?
What is a piecewise function?
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What are points of intersection for two functions?
What are points of intersection for two functions?
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Study Notes
- This is a homework assignment on 1.4 Functions
- The due date for this assignment was on Sunday, February 2, 2025, at 11:59 PM CST
Functions as Equations
- The equation x² + y² = 36, does not represent y as a function of x.
Evaluating Functions
- Given g(t) = 4t² - 3t + 7:
- g(2) = 17
- g(-1) = 14
- g(t + 2) = 4t² + 13t + 17
- Given V(r) = 4⁄3πr³:
- V(3) = 36π
- V(3⁄2) = 9⁄2π
- V(2r) = 32⁄3πr³
- Given f(x) = √(x + 8) + 2:
- f(-8) = 2
- f(1) = 5
- f(x - 8) = √x + 2
- Given q(x) = 1 / (x - 3):
- q(0) = -1/3
- q(3) = undefined
- q(y + 3) = 1/y
Piecewise Functions
- Given f(x) = 2x + 1 if x < 0, and f(x) = 2x + 9 if x ≥ 0:
- f(-1) = -1
- f(0) = 9
- f(2) = 13
- Given f(x) = -5x - 5 if x < -1, and f(x) = x² + 2x - 1 if x ≥ -1:
- f(-3) = 10
- f(-1) = -2
- f(1) = 2
Function Tables
- For f(x) = -x² + 8:
- When x = -2, f(x) = 4
- When x = -1, f(x) = 7
- When x = 0, f(x) = 8
- When x = 1, f(x) = 7
- When x = 2, f(x) = 4
- For h(t) = t + 2⁄t:
- When t = -5, h(t) = -1.5
- When t = -4, h(t) = -0.5
- When t = -3, h(t) = -1⁄3
- When t = -2, h(t) = -1
- When t = -1, h(t) = -3
Zeroes of Functions
- The real values of x for which f(x) = 0:
- f(x) = 35 - 5x, x = 7
- f(x) = 5x - 3, x = 3⁄5
- f(x) = x² - 81, x = 9, -9
- f(x) = x² - 6x - 27, x = 9, -3
- f(x) = x³ - x, x = 0, 1, -1
- f(x) = x³ - x² - 5x + 5, x = 1, √5, -√5
Function Equality
- The values of x for which f(x) = g(x):
- f(x) = x² + 2x + 1, g(x) = 3x + 31, x = 6, -5
- f(x) = x⁴ - 18x², g(x) = 18x², x = 0, 6, -6
- f(x) = √x², g(x) = 10 - x, x = 9
Domains of Functions
- The domain of the function:
- f(x) = 5x² + x - 3 is all real numbers x.
- g(y) = √y + 8 is all real numbers y such that y ≥ -8.
- f(t) = 8 / (t + 8) is all real numbers t except t = -8.
- h(x) = 6 / (x² - 2x) is all real numbers except x = 0 and x = 2.
- f(s) = √(s - 5) is all real numbers s such that s ≥ 5 except s = 1.
- f(x) = (x + 4) / √(x - 10) is all real numbers x such that x > 10.
Applied Functions
- The percent p of prescriptions filled with generic drugs at a large pharmacy chain from 2012 through 2018 is modeled by:
- p(t) = 1.76t + 58.1, 12 ≤ t < 16
- p(t) = 0.91t + 71.2, 16 ≤ t ≤ 18, where t represents the year, with t = 12 corresponding to 2012.
- Percent of prescriptions filled with generic drugs in each year:
- 2012: 79.22%
- 2013: 80.98%
- 2014: 82.74%
- 2015: 84.5%
- 2016: 85.76%
- 2017: 86.67%
- 2018: 87.58%
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