Functions
57 flashcards covering Functions for the ACT Math section.
Functions are a fundamental concept in math that describe a relationship where each input produces exactly one output. Imagine a recipe that turns ingredients into a specific dish—every set of inputs leads to a single result, without exceptions. They can be expressed as equations, like f(x) = x + 2, or through graphs and tables, and they're essential for solving real-world problems, from calculating costs to modeling growth.
On the ACT Math section, functions often show up in questions about evaluating expressions, finding domain and range, graphing linear or quadratic functions, and composing multiple functions. Common traps include confusing functions with non-functions (like circles on graphs) or mishandling variables in equations, so watch for vertical line tests and notation errors. Focus on practicing transformations and word problems to build speed and accuracy.
Remember to double-check your graphs for the vertical line test.
Terms (57)
- 01
Function
A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.
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Domain of a function
The domain of a function is the set of all possible input values for which the function is defined.
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Range of a function
The range of a function is the set of all possible output values that the function can produce.
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Function notation
Function notation expresses a function as f(x), where x is the input and f(x) is the output, allowing for easy evaluation and manipulation.
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Linear function
A linear function is a function whose graph is a straight line, typically written as f(x) = mx + b, where m is the slope and b is the y-intercept.
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Slope of a line
The slope of a line measures its steepness and direction, calculated as the change in y-values divided by the change in x-values for any two points on the line.
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Y-intercept
The y-intercept is the point where a line crosses the y-axis, representing the value of y when x is zero in a linear equation.
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Quadratic function
A quadratic function is a polynomial function of degree two, generally written as f(x) = ax² + bx + c, and its graph is a parabola.
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Vertex of a parabola
The vertex of a parabola is the highest or lowest point on the graph of a quadratic function, occurring at x = -b/(2a) for f(x) = ax² + bx + c.
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Axis of symmetry
The axis of symmetry is the vertical line that divides the parabola of a quadratic function into two mirror-image halves, given by x = -b/(2a).
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Exponential function
An exponential function has the form f(x) = a b^x, where a is a constant, b is the base greater than 0 and not equal to 1, and x is the exponent.
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Logarithmic function
A logarithmic function is the inverse of an exponential function, typically written as f(x) = logb(x), where b is the base and x is greater than 0.
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Base of an exponential
The base of an exponential function is the number that is raised to a power, determining the growth or decay rate in functions like f(x) = a b^x.
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Growth function
A growth function is an exponential function where the base is greater than 1, causing the output to increase as the input increases.
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Decay function
A decay function is an exponential function where the base is between 0 and 1, causing the output to decrease as the input increases.
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Composition of functions
Composition of functions means combining two functions, such as f(g(x)), where the output of the inner function g(x) becomes the input for the outer function f.
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Inverse function
An inverse function reverses the operation of the original function, so if f(x) = y, then f⁻¹(y) = x, provided the function is one-to-one.
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One-to-one function
A one-to-one function is a function where each output corresponds to exactly one input, allowing it to have an inverse.
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Horizontal line test
The horizontal line test determines if a function is one-to-one by checking if any horizontal line intersects the graph more than once.
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Vertical line test
The vertical line test determines if a relation is a function by checking if any vertical line intersects the graph more than once.
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Translation of a function
Translation of a function shifts its graph horizontally or vertically, such as f(x) + k shifts it up by k units or f(x - h) shifts it right by h units.
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Reflection of a function
Reflection of a function flips its graph over a line, like f(-x) reflects over the y-axis or -f(x) reflects over the x-axis.
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Stretching or compressing a function
Stretching or compressing a function alters its graph vertically or horizontally, such as af(x) stretches vertically by factor a if a > 1.
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Even function
An even function satisfies f(-x) = f(x) for all x in the domain, meaning its graph is symmetric with respect to the y-axis.
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Odd function
An odd function satisfies f(-x) = -f(x) for all x in the domain, meaning its graph is symmetric with respect to the origin.
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Piecewise function
A piecewise function is defined by different expressions over different intervals of its domain, such as f(x) = x if x > 0 and f(x) = -x if x ≤ 0.
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Absolute value function
The absolute value function is f(x) = |x|, which outputs the non-negative value of x, and its graph is a V-shape with the vertex at the origin.
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Rational function
A rational function is a function that is the ratio of two polynomials, such as f(x) = (x^2 + 1)/(x - 2), and it may have asymptotes.
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Asymptote of a function
An asymptote is a line that a graph approaches but never touches, common in rational and exponential functions, such as vertical or horizontal lines.
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End behavior of a function
End behavior describes what happens to the output of a function as the input approaches positive or negative infinity, like polynomials going to infinity.
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Zeros of a function
The zeros of a function are the values of x for which f(x) = 0, also known as the roots or x-intercepts of the graph.
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X-intercept
An x-intercept is a point where the graph of a function crosses the x-axis, meaning y = 0 at that point.
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Increasing function
An increasing function is one where the output values rise as the input values increase, meaning if x1 < x2, then f(x1) < f(x2).
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Decreasing function
A decreasing function is one where the output values fall as the input values increase, meaning if x1 < x2, then f(x1) > f(x2).
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Maximum value of a function
The maximum value of a function is the highest output value it reaches, often at the vertex for quadratics or in a specified interval.
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Minimum value of a function
The minimum value of a function is the lowest output value it reaches, such as the vertex of an upward-opening parabola.
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Parent function
A parent function is the simplest form of a particular type of function, like f(x) = x for linear or f(x) = x^2 for quadratic, from which others are derived.
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Function evaluation
Function evaluation means substituting a specific value into a function to find the output, such as finding f(2) for f(x) = x + 3.
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Solving for x in a function
Solving for x in a function involves finding the input values that produce a given output, such as solving f(x) = k for x.
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Graph of a function
The graph of a function is a visual representation on the coordinate plane, showing all points (x, f(x)) for x in the domain.
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Domain restrictions
Domain restrictions are conditions that limit the inputs of a function, such as excluding values that make a denominator zero in rational functions.
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Range restrictions
Range restrictions are limitations on the possible outputs of a function, often determined by analyzing its graph or equation.
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Function tables
Function tables are organized lists of input and output values for a function, helping to identify patterns or graph the function.
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Real-world applications of functions
Functions model real-world situations, like distance as a function of time in motion problems or cost as a function of quantity in economics.
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Cost function
A cost function represents total cost as a function of the number of items produced, often including fixed and variable costs.
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Revenue function
A revenue function expresses total revenue as a function of the number of items sold, typically as price per item times quantity.
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Break-even point
The break-even point is the value where a cost function equals a revenue function, indicating no profit or loss.
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Arithmetic sequence as a function
An arithmetic sequence can be viewed as a linear function where each term is the previous term plus a constant difference.
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Geometric sequence as a function
A geometric sequence can be represented as an exponential function where each term is the previous term multiplied by a constant ratio.
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Quadratic formula
The quadratic formula solves for x in ax² + bx + c = 0, given by x = [-b ± sqrt(b² - 4ac)] / (2a), finding the roots of a quadratic function.
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Factoring quadratics
Factoring quadratics involves rewriting ax² + bx + c as (dx + e)(fx + g) to solve for roots or simplify the function.
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Completing the square
Completing the square rewrites a quadratic function in vertex form by adding and subtracting a constant, useful for finding the vertex.
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Vertex form of a quadratic
Vertex form of a quadratic is f(x) = a(x - h)² + k, where (h, k) is the vertex, making it easy to identify the maximum or minimum.
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Standard form of a quadratic
Standard form of a quadratic is f(x) = ax² + bx + c, which is used to identify coefficients for graphing or solving.
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Slope-intercept form
Slope-intercept form is y = mx + b, a linear function format that directly shows the slope and y-intercept.
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Point-slope form
Point-slope form is y - y1 = m(x - x1), used to write the equation of a line when a point and slope are known.
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Systems of linear functions
Systems of linear functions involve solving two or more equations simultaneously to find intersection points, often representing real-world constraints.