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Functions

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What is a function in mathematics?

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A function $f$ is a rule that assigns to each element $x$ in a set $A$ exactly one element $y$ in a set $B$ , written as $f: A \to B$ or $y = f(x)$ .

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What is a function in mathematics?

A function $f$ is a rule that assigns to each element $x$ in a set $A$ exactly one element $y$ in a set $B$, written as $f: A \to B$ or $y = f(x)$.

What are the domain and codomain of a function?

The domain $A$ is the set of all possible input values, while the codomain $B$ is the set of all possible output values.

What is the range of a function?

The range is the set of all $y$ values actually produced by the function.

How is the graph of a function defined?

The graph of a function is the set of all points $(x, y)$ such that $y = f(x)$.

What is an inverse function?

An inverse function $f^{-1}$ undoes the action of the function $f$, satisfying $f^{-1}(f(x)) = x$.

How do you find the inverse of a function algebraically?

1. Replace $f(x)$ with $y$; 2. Swap $x$ and $y$; 3. Solve for $y$; 4. Replace $y$ with $f^{-1}(x)$.

What is a composite function?

A composite function $(f \circ g)(x)$ is defined as $f(g(x))$, applying $g$ first and then $f$.

What is the general form of a quadratic function?

A quadratic function has the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.

What are the key features of a quadratic function?

Vertex: $(-\frac{b}{2a}, f(-\frac{b}{2a}))$; Axis of symmetry: $x = -\frac{b}{2a}$; $y$-intercept: $(0, c)$.

What is a rational function?

A rational function is a ratio of polynomials, expressed as $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

What are the key features of a rational function?

Zeros: Where $P(x) = 0$; Vertical asymptotes: Where $Q(x) = 0$; Horizontal asymptote: Compare degrees of $P(x)$ and $Q(x)$.

What is the definition of an exponential function?

An exponential function is of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

What is the definition of a logarithmic function?

A logarithmic function is of the form $f(x) = \log_a(x)$, where $a > 0$ and $a \neq 1$.

What are the transformations of graphs?

Transformations include vertical/horizontal shifts, vertical/horizontal stretches/compressions, and reflections.

What is the Fundamental Theorem of Algebra?

A polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicity).

What defines odd and even functions?

Odd function: $f(-x) = -f(x)$; Even function: $f(-x) = f(x)$.

What is the modulus function?

The modulus function is defined as $|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}$.

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1 / 17

What is a function in mathematics?

Flip

Spacebar

A function $f$ is a rule that assigns to each element $x$ in a set $A$ exactly one element $y$ in a set $B$ , written as $f: A \to B$ or $y = f(x)$ .

Flip

Spacebar

What is a function in mathematics?

A function $f$ is a rule that assigns to each element $x$ in a set $A$ exactly one element $y$ in a set $B$, written as $f: A \to B$ or $y = f(x)$.

What are the domain and codomain of a function?

The domain $A$ is the set of all possible input values, while the codomain $B$ is the set of all possible output values.

What is the range of a function?

The range is the set of all $y$ values actually produced by the function.

How is the graph of a function defined?

The graph of a function is the set of all points $(x, y)$ such that $y = f(x)$.

What is an inverse function?

An inverse function $f^{-1}$ undoes the action of the function $f$, satisfying $f^{-1}(f(x)) = x$.

How do you find the inverse of a function algebraically?

1. Replace $f(x)$ with $y$; 2. Swap $x$ and $y$; 3. Solve for $y$; 4. Replace $y$ with $f^{-1}(x)$.

What is a composite function?

A composite function $(f \circ g)(x)$ is defined as $f(g(x))$, applying $g$ first and then $f$.

What is the general form of a quadratic function?

A quadratic function has the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.

What are the key features of a quadratic function?

Vertex: $(-\frac{b}{2a}, f(-\frac{b}{2a}))$; Axis of symmetry: $x = -\frac{b}{2a}$; $y$-intercept: $(0, c)$.

What is a rational function?

A rational function is a ratio of polynomials, expressed as $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

What are the key features of a rational function?

Zeros: Where $P(x) = 0$; Vertical asymptotes: Where $Q(x) = 0$; Horizontal asymptote: Compare degrees of $P(x)$ and $Q(x)$.

What is the definition of an exponential function?

An exponential function is of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

What is the definition of a logarithmic function?

A logarithmic function is of the form $f(x) = \log_a(x)$, where $a > 0$ and $a \neq 1$.

What are the transformations of graphs?

Transformations include vertical/horizontal shifts, vertical/horizontal stretches/compressions, and reflections.

What is the Fundamental Theorem of Algebra?

A polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicity).

What defines odd and even functions?

Odd function: $f(-x) = -f(x)$; Even function: $f(-x) = f(x)$.

What is the modulus function?

The modulus function is defined as $|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}$.