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Precalculus Chapters 1.1, 1.2, 1.3, 1.6, 1.7, 1.8, and 1.10 Key Concepts

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Termini in questo insieme (20)

  • What is a function?

    A function is a relation where each input has exactly one output.

  • How do you determine if a graph represents a function?

    Use the vertical line test: if any vertical line crosses the graph more than once, it is not a function.

  • What is the domain of a function?

    The domain is the set of all possible input values (x-values) for which the function is defined.

  • What is the range of a function?

    The range is the set of all possible output values (y-values) of the function.

  • How do you find the inverse of a function?

    Swap the roles of x and y, then solve for y to find the inverse function.

  • What is the difference between even and odd functions?

    Even functions satisfy \(f(-x)=f(x)\); odd functions satisfy \(f(-x)=-f(x)\).

  • What is a piecewise function?

    A piecewise function is defined by different expressions over different parts of its domain.

  • How do you perform function composition?

    Function composition \((f \circ g)(x)\) means applying g first, then f: \(f(g(x))\).

  • What is the standard form of a linear function?

    The standard form is \(Ax + By = C\), where A, B, and C are constants.

  • How do you find the slope of a line given two points?

    Slope m is \(\frac{y_2 - y_1}{x_2 - x_1}\).

  • What is the slope-intercept form of a line?

    Slope-intercept form is \(y = mx + b\), where m is slope and b is y-intercept.

  • How do you graph a linear inequality?

    Graph the boundary line (solid if ≤ or ≥, dashed if < or >), then shade the side that satisfies the inequality.

  • What is the difference between a relation and a function?

    A relation is any set of ordered pairs; a function assigns exactly one output to each input.

  • What is the intercept form of a linear equation?

    Intercept form is \(\frac{x}{a} + \frac{y}{b} = 1\), where a and b are x- and y-intercepts.

  • How do you find the x- and y-intercepts of a function?

    Set y = 0 to find x-intercept; set x = 0 to find y-intercept.

  • What is the domain of a rational function?

    The domain excludes values that make the denominator zero.

  • How do you simplify complex fractions?

    Find a common denominator for numerator and denominator, then rewrite and simplify.

  • What is the difference quotient formula?

    The difference quotient is \(\frac{f(x+h) - f(x)}{h}\), used to find average rate of change.

  • What is the significance of the difference quotient?

    It represents the average rate of change of a function over an interval of length h.

  • How do you determine if a function is one-to-one?

    A function is one-to-one if each output corresponds to exactly one input; use the horizontal line test.