Here are definitions and illustrations for the vocabulary terms provided:
Initial Value
Definition: The initial value of a function is the value of the function at the starting point, often when \( x = 0 \). In a graph, this is the point where the graph intersects the y-axis.
Illustration: If we have a linear function expressed as \( f(x) = 2x + 3 \), the initial value would be \( f(0) = 2(0) + 3 = 3 \). The graph of this function crosses the y-axis at the point (0, 3).
X-Intercept
Definition: The x-intercept of a function is the point at which the graph of the function crosses the x-axis. It occurs when the output value (y-value) is equal to zero.
Illustration: For the equation \( f(x) = x^2 - 4 \), we can find the x-intercepts by setting \( f(x) = 0 \): \[ x^2 - 4 = 0 \implies (x-2)(x+2) = 0 \implies x = 2 \text{ and } x = -2 \] The x-intercepts are at the points (2, 0) and (-2, 0) on the graph.
Domain/Range
Definitions:
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Domain: The domain of a function is the complete set of possible values of the independent variable (usually \( x \)). It describes the inputs for which the function is defined.
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Range: The range of a function is the complete set of possible output values (usually \( y \)) that the function can produce based on its domain.
Illustration: Consider the function \( f(x) = \sqrt{x} \).
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Domain: The expression \( \sqrt{x} \) is defined for \( x \geq 0 \) (i.e., it cannot take negative values). Thus, the domain is \( [0, \infty) \).
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Range: Since the square root function outputs non-negative values, the range is also \( [0, \infty) \).
In summary:
- Initial value: \( f(0) \) on the y-axis.
- X-intercept: Points where \( f(x) = 0 \) on the x-axis.
- Domain: Set of possible x-values; Range: Set of possible y-values.
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