Functions on the GMAT
52 flashcards covering Functions on the GMAT for the GMAT Quantitative section.
Functions are mathematical rules that relate inputs to outputs in a predictable way. For example, if you have a function like f(x) = 2x + 3, it means that for any number you plug in for x, you'll get a specific result as the output. This concept is fundamental in algebra and helps model real-world scenarios, such as calculating costs or growth rates. On the GMAT, understanding functions is essential because they appear in various quantitative problems, testing your ability to evaluate expressions, solve equations, and interpret graphs.
In the Quantitative section, functions often show up in problem-solving and data-sufficiency questions, where you might need to find values, determine domains and ranges, or compose functions. Common traps include misreading function notation, overlooking restrictions like undefined values, or confusing variables in composite functions. Focus on mastering basic operations, such as substitution and graphing, to handle these efficiently and avoid careless errors.
Practice evaluating functions quickly to build confidence.
Terms (52)
- 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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Function Notation
Function notation expresses a function using f(x), where x is the input and f(x) is the output, allowing for clear evaluation and manipulation.
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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, often restricted by denominators or square roots.
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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 for inputs in its domain.
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Linear Function
A linear function is a function of the form f(x) = mx + b, where m is the slope and b is the y-intercept, representing a straight line on a graph.
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Slope
Slope measures the steepness of a line in a linear function, calculated as the change in y-values divided by the change in x-values, indicating rise over run.
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Y-Intercept
The y-intercept is the value of y when x is zero in a linear function, representing the point where the line crosses the y-axis.
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Quadratic Function
A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0, typically graphing as a parabola.
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Vertex of a Quadratic
The vertex of a quadratic function is the highest or lowest point on its parabola, found at x = -b/(2a) in the standard form.
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Axis of Symmetry
The axis of symmetry in a quadratic function is the vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
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Exponential Function
An exponential function is a function of the form f(x) = a b^x, where a and b are constants, b > 0 and b ≠ 1, used to model growth or decay.
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Logarithmic Function
A logarithmic function is the inverse of an exponential function, typically f(x) = logb(x), where b is the base and x > 0.
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Base of a Logarithm
The base of a logarithm is the number that is raised to a power in the corresponding exponential equation, such as b in logb(x) = y.
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Composite Function
A composite function is formed by applying one function to the result of another, denoted as f(g(x)), where the output of g(x) becomes the input for 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, necessary for the function to have an inverse.
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Even Function
An even function satisfies f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis.
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Odd Function
An odd function satisfies f(-x) = -f(x) for all x in its domain, meaning its graph is symmetric about the origin.
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Piecewise Function
A piecewise function is defined by different expressions over different intervals of its domain, allowing it to behave differently in various regions.
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Absolute Value Function
The absolute value function, f(x) = |x|, outputs the non-negative value of x, graphing as a V-shape with the vertex at the origin.
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Evaluating Functions
Evaluating a function means substituting a specific value into the function to find the output, such as finding f(2) for f(x) = x + 3.
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Adding Functions
Adding functions means combining two functions by adding their outputs, so (f + g)(x) = f(x) + g(x) for the same input x.
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Multiplying Functions
Multiplying functions means combining two functions by multiplying their outputs, so (f g)(x) = f(x) g(x) for the same input x.
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Function Composition
Function composition applies one function to the output of another, calculated as f(g(x)), ensuring the output of the inner function is in the domain of the outer.
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Finding the Inverse
Finding the inverse of a function involves swapping x and y and solving for y, then verifying it undoes the original function.
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Domain of Composite Function
The domain of a composite function f(g(x)) is the set of x values in g's domain such that g(x) is in f's domain.
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Graph of a Function
The graph of a function is a visual representation of all input-output pairs, where each x-value corresponds to exactly one y-value.
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Intercepts of a Function
Intercepts of a function are the points where its graph crosses the axes, with x-intercepts at y=0 and y-intercepts at x=0.
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Symmetry in Functions
Symmetry in functions refers to patterns like even functions being symmetric about the y-axis or odd functions about the origin.
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Maximum and Minimum Values
Maximum and minimum values of a function are the highest and lowest points in a given interval, often found at critical points or endpoints.
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Increasing and Decreasing Intervals
Increasing intervals are where a function's values rise as x increases, and decreasing intervals are where they fall.
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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 Asymptotes
Vertical asymptotes are vertical lines that a function approaches but never touches, occurring where the function is undefined, like at denominators of zero.
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Horizontal Asymptotes
Horizontal asymptotes are horizontal lines that a function approaches as x goes to positive or negative infinity, indicating long-term behavior.
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Solving Functional Equations
Solving functional equations involves finding values of x that satisfy an equation involving functions, such as f(x) = f(2x).
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Common Mistake: Domain Errors
A common mistake with domain errors is forgetting to exclude values that make denominators zero or expressions under square roots negative.
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Example: Linear Function Evaluation
For f(x) = 2x + 1, evaluating at x=3 gives f(3) = 7, showing how to plug in values to find outputs.
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Example: Quadratic Vertex
For f(x) = x^2 - 4x + 3, the vertex is at (2, -1), illustrating how to find the minimum point of a parabola.
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Strategy: Simplifying Expressions
A strategy for simplifying expressions with functions is to substitute and combine like terms, ensuring to follow order of operations.
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Function as a Machine
Viewing a function as a machine means it takes an input, processes it according to a rule, and produces a single output, emphasizing the one-to-one mapping.
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Dependent and Independent Variables
In a function, the independent variable is the input, like x, while the dependent variable is the output, like f(x), which depends on the input.
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Function Tables
Function tables organize input and output values in a chart, helping to visualize patterns and evaluate functions at multiple points.
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Word Problems with Functions
Word problems with functions translate real-world scenarios into function form, such as cost functions, to solve for unknowns.
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Cost Functions
Cost functions model total expenses as a function of quantity, often linear like C(q) = fixed cost + variable cost per unit times q.
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Revenue Functions
Revenue functions express total income as a function of quantity sold, typically R(q) = price per unit times q.
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Profit Functions
Profit functions calculate net gain as revenue minus costs, so P(q) = R(q) - C(q), used to find break-even points.
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Break-Even Point
The break-even point is the quantity where profit is zero, found by solving P(q) = 0 in a profit function.
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Exponential Growth Model
An exponential growth model is a function like f(x) = a b^x where b > 1, used to represent increasing quantities over time.
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Logarithmic Equations
Logarithmic equations involve logs, solved by converting to exponential form or using log properties to isolate variables.
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Change of Base Formula
The change of base formula for logarithms is logb(a) = logc(a) / logc(b), allowing calculation with any base.
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Properties of Logarithms
Properties of logarithms include the product rule, quotient rule, and power rule, which simplify expressions like log(ab) = log(a) + log(b).
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Natural Exponential Function
The natural exponential function is f(x) = e^x, where e is approximately 2.718, used in growth and decay models.