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Understanding Inverse Functions: Their Properties, Examples, and Applications

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helmut
7 diapositives · 229 Views · Aug 13, 2014

This text provides an in-depth discussion on inverse functions, including the necessary conditions for their existence, how to find inverses algebraically, and graphical representations demonstrating their symmetry about the line ( y = x ). It also covers practical examples related to rental pricing and tenant numbers, illustrating the application of inverse functions in real-world scenarios. The content includes key concepts such as the vertical line test (VLT) and horizontal line test (HLT), along with polynomial function properties and end behavior considerations.

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  1. 2.7 Inverse Functions • Pg. 136 #10 – 34 evenPg. 150 #45 – 49 all • #9 [-2, 7) #27 No Solutions • #11 (-2, 8) #29 x = -1.00, 0.50 • #13 (1, 2) #31 x = 2.23 • #15 (-∞, -6]U[-2, ∞) #33 x = • #17 (-2/5, 4/5) #35 x = 0.85 • #19 x = #42 (a) 450 – 15x = rent • #21 x = (b) 1900+20x =tenants • #23 x = -0.44, 1.69 #43 [0, 30) • #25 x = 2.11 #44 x = 0, rent = $450

  2. 2.7 Inverse Functions Pg. 50 #42 – 44 #42 – Money and people stick together. That’s why there are two equations: 450 – 15x = rent 1900 + 20x = tenants #43 – Revenue is the number of tenants times the cost of rent, or those two equations multiplied. So, R = (450 – 15x)(1900 + 20x). Set each piece equal to zero and you know you’re boundaries. x = 0 and x = 30, so [0, 30) because you don’t want R = 0. #44 – Graph R in your calculator and find the maximum. It will occur when x = 0 and R = 450. • A large apartment rental company has 2500 units available, and 1900 are currently rented at an average of $450/mo. A market survey indicates that each $15 decrease in average monthly rent will result in 20 new tenants.

  3. 2.7 Inverse Functions Inverse Relations Inverse Functions In order for an inverse function to exist, first you must be dealing with a function and that function must pass the VLT and the HLT. Functions and Inverse Functions can be composed together to prove they are inverses of each other. Their result will always be x. • The point (a, b) is in the relation R if, and only if, (b, a) is in the relation R-1. • Graphically, an inverse is a reflection of the original graph over the line y = x.

  4. 2.7 Inverse Functions Examples Find the inverse of f(x) = -x3 algebraically. Graph the original equation and the inverse along with the line y = x to show it has the proper symmetry. • Find the inverse of y = ½ x – 3 algebraically. • Graph the original equation and the inverse along with the line y = x to show it has the proper symmetry.

  5. 2.7 Inverse Functions Inverse Functions Show that g(x) = will have an inverse function. Find the inverse function and state its domain and range. Prove that the two are actually inverses. Show that h(x) = x3 – 5xwill have an inverse function. • Show that f(x) = will have an inverse function. • Find the inverse function and state its domain and range. • Prove that the two are actually inverses.

  6. 3.1 Graphs of Polynomial Functions Definition State whether the following are polynomials. If so, state the degree. • A polynomial function is one that can be written in the form:where n is a nonnegative integer and the coefficients are real numbers. If the leading coefficient is not zero, then n is the degree of the polynomial.

  7. 3.1 Graphs of Polynomial Functions End Behavior Number of “Bumps” The number of “bumps” a graph may have is no more than one less than the degree. The number of zeros a graph may have is no more than the number of the degree. • End behavior is determined by the degree and the leading coefficient. • Create Chart.