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Questions and Answers
Consider the function $f(x) = x^3 - x$. Which transformation will result in a function symmetric with respect to the y-axis?
Consider the function $f(x) = x^3 - x$. Which transformation will result in a function symmetric with respect to the y-axis?
- Replacing $x$ with $-x$ and multiplying the function by $-1$.
- Adding a constant value to the function, $f(x) + c$.
- Replacing $x$ with $-x$. (correct)
- Multiplying the function by $-1$.
A rational function has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 1$. Which of the following functions could represent this rational function?
A rational function has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 1$. Which of the following functions could represent this rational function?
- $f(x) = \frac{2x}{x - 1}$
- $f(x) = \frac{x - 2}{x + 1}$
- $f(x) = \frac{x}{x + 2}$
- $f(x) = \frac{x + 1}{x - 2}$ (correct)
How does the graph of $f(x) = |x|$ differ from the graph of $g(x) = -|x|$?
How does the graph of $f(x) = |x|$ differ from the graph of $g(x) = -|x|$?
- g(x) is shifted horizontally to the left compared to f(x).
- g(x) is a reflection of f(x) over the y-axis.
- g(x) is shifted vertically downward compared to f(x).
- g(x) is a reflection of f(x) over the x-axis. (correct)
If the function $f(x)$ is even and $f(2) = 5$, what is the value of $f(-2)$?
If the function $f(x)$ is even and $f(2) = 5$, what is the value of $f(-2)$?
Which transformation applied to $f(x) = x^2$ will result in the function $g(x) = -x^2 + 3$?
Which transformation applied to $f(x) = x^2$ will result in the function $g(x) = -x^2 + 3$?
Which of the following transformations would result in a vertical compression of the function $f(x)$?
Which of the following transformations would result in a vertical compression of the function $f(x)$?
Given two functions, $f(x)$ and $g(x)$, which of the following statements is generally true regarding their composition?
Given two functions, $f(x)$ and $g(x)$, which of the following statements is generally true regarding their composition?
What condition must be met for a relation to be considered a function?
What condition must be met for a relation to be considered a function?
Consider the function $h(x) = \frac{f(x)}{g(x)}$. What must be true about the domains of $f(x)$ and $g(x)$ to determine the domain of $h(x)$?
Consider the function $h(x) = \frac{f(x)}{g(x)}$. What must be true about the domains of $f(x)$ and $g(x)$ to determine the domain of $h(x)$?
How does the graph of $f(x) = x^2$ change if transformed to $g(x) = (x - 2)^2 + 3$?
How does the graph of $f(x) = x^2$ change if transformed to $g(x) = (x - 2)^2 + 3$?
Which of the following statements accurately describes the end behavior of a function?
Which of the following statements accurately describes the end behavior of a function?
Which of the following equations represents an odd function?
Which of the following equations represents an odd function?
Which of the following operations on functions requires excluding x-values where $g(x) = 0$ when $g(x)$ is involved?
Which of the following operations on functions requires excluding x-values where $g(x) = 0$ when $g(x)$ is involved?
Flashcards
Reflection over x-axis
Reflection over x-axis
A transformation where the graph is flipped over the x-axis. Achieved by multiplying the function by -1.
Reflection over y-axis
Reflection over y-axis
A transformation where the graph is flipped over the y-axis. Achieved by replacing x with -x.
Even Function
Even Function
A function where f(x) = f(-x). It is symmetric with respect to the y-axis.
Odd Function
Odd Function
A function where f(-x) = -f(x). Symmetric with respect to the origin.
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Vertical Asymptote
Vertical Asymptote
A line that a function approaches but does not touch. Vertical asymptotes often occur where the denominator of a rational function is zero.
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What is a function?
What is a function?
A relation where each input (x) has only one output (y).
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What is the domain?
What is the domain?
All possible input values (x-values) for which a function is defined.
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What is the range?
What is the range?
All possible output values (y-values) that a function can produce.
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What are intercepts?
What are intercepts?
Points where the function's graph crosses the x-axis or y-axis.
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What is an even function?
What is an even function?
Symmetry about the y-axis, where f(-x) = f(x).
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What is function composition?
What is function composition?
Applying one function to the result of another: f(g(x)).
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What is a vertical shift?
What is a vertical shift?
Shifting a function's graph vertically by adding a constant: f(x) + c.
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Horizontal stretch/compression
Horizontal stretch/compression
Replacing x with (cx) in the function, f(cx). Compresses if |c| > 1, stretches if 0 < |c| < 1.
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- A function is a relation where each input has exactly one output.
Function Properties
- Domain refers to all possible input values (x-values) for which the function is defined.
- Range refers to all possible output values (y-values) that the function can produce.
- Intercepts are points where the function's graph intersects the x-axis (x-intercept) or the y-axis (y-intercept).
- Symmetry can be even (symmetric about the y-axis, f(-x) = f(x)), odd (symmetric about the origin, f(-x) = -f(x)), or neither.
- Intervals of increase and decrease: where the function's values are going up or down as x increases.
- Maximum and minimum values: the highest and lowest points of the function, either locally or globally.
- End behavior: describes what happens to the function’s values as x approaches positive or negative infinity.
- Continuity: A function is continuous if its graph can be drawn without lifting the pen, meaning there are no breaks, holes, or asymptotes.
- Discontinuity: Points where a function is not continuous. Can be removable (hole), jump, or infinite (asymptote).
Function Operations
- Addition: (f + g)(x) = f(x) + g(x)
- Subtraction: (f - g)(x) = f(x) - g(x)
- Multiplication: (f * g)(x) = f(x) * g(x)
- Division: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
- Scalar Multiplication: (c * f)(x) = c * f(x), where c is a constant.
- The domain of the resulting function consists of x-values that are in the domains of f and g.
- For division, we also exclude any x-values where g(x) = 0.
Function Composition
- Composition: (f ∘ g)(x) = f(g(x)), meaning apply g first, then apply f to the result.
- To evaluate, first find g(x) and then substitute that expression into f(x).
- The domain of (f ∘ g)(x) is the set of all x in the domain of g such that g(x) is in the domain of f.
- Composition is not commutative: f(g(x)) is generally not equal to g(f(x)).
Graphing Functions
- Plotting points: calculate several points (x, f(x)) and plot them on a coordinate plane.
- Transformations: altering the graph of a function by shifting, stretching, compressing, or reflecting.
- Vertical shift: adding a constant to the function, f(x) + c (up if c > 0, down if c < 0).
- Horizontal shift: replacing x with (x - c) in the function, f(x - c) (right if c > 0, left if c < 0).
- Vertical stretch/compression: multiplying the function by a constant, c * f(x) (stretch if |c| > 1, compress if 0 < |c| < 1).
- Horizontal stretch/compression: replacing x with (c * x) in the function, f(cx) (compress if |c| > 1, stretch if 0 < |c| < 1).
- Reflection over the x-axis: multiplying the function by -1, -f(x).
- Reflection over the y-axis: replacing x with -x, f(-x).
- Symmetry:
- Even functions are symmetric with respect to the y-axis.
- Odd functions are symmetric with respect to the origin.
- Asymptotes:
- Vertical asymptotes occur where the function approaches infinity (often where the denominator of a rational function is zero).
- Horizontal asymptotes describe the function's behavior as x approaches infinity or negative infinity. Found by examining the limits as x goes to ±∞.
- Key functions to know:
- Linear functions: f(x) = mx + b
- Quadratic functions: f(x) = ax^2 + bx + c
- Polynomial functions
- Rational functions: p(x)/q(x) where p(x) and q(x) are polynomials.
- Exponential functions: f(x) = a^x
- Logarithmic functions: f(x) = log_a(x)
- Trigonometric functions: sin(x), cos(x), tan(x)
- Absolute value function: f(x) = |x|
- Square root function: f(x) = √x
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