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§ 0.2. Some Important Functions. Section Outline. Linear Equations Applications of Linear Functions Piece-Wise Functions Quadratic Functions Polynomial Functions Rational Functions Power Functions Absolute Value Function. Linear Equations. Linear Equations. CONTINUED.

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§ 0.2

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  1. §0.2 Some Important Functions

  2. Section Outline • Linear Equations • Applications of Linear Functions • Piece-Wise Functions • Quadratic Functions • Polynomial Functions • Rational Functions • Power Functions • Absolute Value Function

  3. Linear Equations

  4. Linear Equations CONTINUED

  5. Applications of Linear Functions EXAMPLE • (Enzyme Kinetics) In biochemistry, such as in the study of enzyme kinetics, one encounters a linear function of the form , where K and V are constants. • If f (x) = 0.2x + 50, find K and V so that f (x) may be written in the form, . • Find the x-intercept and y-intercept of the line in terms of K and V. SOLUTION (a) Since the number 50 in the equation f (x) = 0.2x + 50 is in place of the term 1/V (from the original function), we know the following. 50 = 1/V Explained above. 50V = 1 Multiply both sides by V. Divide both sides by 50. V = 0.02 Now that we know what V is, we can determine K. Since the number 0.2 in the equation f (x) = 0.2x + 50 is in place of K/V (from the original function), we know the following.

  6. Applications of Linear Functions CONTINUED 0.2 = K/V Explained above. 0.2V = K Multiply both sides by V. Replace V with 0.02. 0.2(0.02) = K 0.004 = K Multiply. Therefore, in the equation f (x) = 0.2x + 50, K = 0.004 and V = 0.02. (b) To find the x-intercept of the original function, replace f (x) with 0. This is the original function. Replace f (x) with 0. Solve for x by first subtracting 1/V from both sides.

  7. Applications of Linear Functions CONTINUED Multiply both sides by V/K. Simplify. Therefore, the x-intercept is -1/K. To find the y-intercept of the original function, we recognize that this equation is in the form y = mx + b. Therefore we know that 1/V is the y-intercept.

  8. Piece-Wise Functions EXAMPLE Sketch the graph of the following function . SOLUTION We graph the function f (x) = 1 + x only for those values of x that are less than or equal to 3. Notice that for all values of x greater than 3, there is no line.

  9. Piece-Wise Functions CONTINUED Now we graph the function f (x) = 4 only for those values of x that are greater than 3. Notice that for all values of x less than or equal to 3, there is no line.

  10. Piece-Wise Functions CONTINUED Now we graph both functions on the same set of axes.

  11. Quadratic Functions

  12. Polynomial Functions

  13. Rational Functions

  14. Power Functions

  15. Absolute Value Function

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