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College Algebra: Graphs, Intercepts, Domain, and Range

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  • What is the x-intercept of a graph?

    The x-intercept is the point(s) where the graph crosses the x-axis, where \(y=0\) or \(f(x)=0\).

  • How do you determine the domain of a function from its graph?

    The domain is the set of all possible x-values for which the function is defined, found by looking at the leftmost and rightmost points on the graph along the x-axis.

  • How is the domain written in set-builder notation?

    Domain is written as \(\{x\mid \text{condition on } x\}\), for example, \(\{x\mid a \leq x \leq b\}\) where a and b are endpoints.

  • How do you find the x-value where \(f(x)=5\) from a graph?

    Locate y = 5 on the vertical axis, draw a horizontal line at y = 5, and find the x-coordinate(s) where this line intersects the graph.

  • How do you evaluate \(f(1)\) using a graph?

    Find x = 1 on the x-axis, move vertically to the graph, and read the corresponding y-value, which is \(f(1)\).

  • What is the range of a function from its graph?

    The range is the set of all y-values that the function attains, found by looking at the lowest and highest points on the graph.

  • How do you determine if endpoints are included in domain or range intervals?

    Check if endpoints have closed circles or solid lines (included, use ≤ or ≥) or open circles/gaps (excluded, use < or >).

  • How do you plot the point \(\left(-\frac{7}{2}, \frac{9}{2}\right)\) on the Cartesian plane?

    Convert to decimals if needed, move left 3.5 units on x-axis, then up 4.5 units on y-axis, and mark the point.

  • What is the y-intercept of a graph?

    The y-intercept is the point where the graph crosses the y-axis, where \(x=0\).

  • How do you find the x-intercepts and y-intercept from a graph?

    Find points where the graph crosses the x-axis (x-intercepts) and the point where it crosses the y-axis (y-intercept).

  • What does the notation \(\{x\mid a \leq x \leq b\}\) represent?

    It represents the set of all x-values between a and b, including the endpoints a and b.

  • What does an open circle on a graph endpoint indicate about the domain or range?

    An open circle means the endpoint is excluded from the domain or range, so use < or > in notation.

  • What does a closed circle on a graph endpoint indicate about the domain or range?

    A closed circle means the endpoint is included in the domain or range, so use ≤ or ≥ in notation.

  • How do you write the domain if the function is defined for all real numbers?

    The domain is \(\{x\mid -\infty < x < \infty\}\) or simply all real numbers.

  • What is the significance of the point where the graph crosses the x-axis?

    It represents the root or zero of the function where \(f(x)=0\).

  • How can you identify the range from a graph with a maximum value?

    The range includes all y-values up to the maximum y-value, including it if the graph has a closed circle or solid line at that point.

  • How can you identify the range from a graph with a minimum value?

    The range includes all y-values starting from the minimum y-value, including it if the graph has a closed circle or solid line at that point.

  • What does it mean if a function's graph has gaps or breaks?

    The function is not defined at those x-values, so those points are excluded from the domain.

  • How do you interpret the function value \(f(c)\) from a graph?

    It is the y-coordinate of the point on the graph where x = c.

  • What is the Cartesian coordinate system?

    A plane defined by a horizontal x-axis and a vertical y-axis used to plot points as (x, y).