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# Math 1450 Chapter 4 Review

 MTH 1450 Chapter 4 Review Practice Problems 1 Inverse functions Tool: Intuitively, inverse functions help us solve equations. For example if we have the equation x3=7 , we apply cube root to both sides. The cube root cancels with the cube on x so we can solve Tool: Inverse functions are defined by their composition properties, so g( x) is the inverse of f ( x ) iff and If g is the inverse of f , we name it f -1 (this does NOT mean ). Tool: The graphs of f and f -1 are reflections about the line y=x , which means if you take points on one graph, switch the x , and y coordinates, we obtain points on the other graph. When we switch x and y , we also exchange domains and ranges. NOTE: not all functions have graphs that are inverses. Tool: A function f has an inverse that is also a function iff a every horizontal line passes through its graph in at most one point. (If you switch x and y , horizontal line test becomes vertical line test for the inverse graph. Any function passing the horizontal line test is said to be oneto- one, i.e., every output comes from exactly one input. Tool: Finding inverse given equation y= f ( x) . Just switch x and y (replace one with the other where you find them). Then solve for the new y= f -1( x ) . NOTE: In applications we don't always use x and y . In general we need to solve for input variable so it becomes the new output variable, the former output becomes input.    2 Exponential functions Tool: Exponential functions appear in the form y=bx. The base, b must be positive. If b >1 the function is increasing. If b <1 , the function is decreasing. Exponential functions have horizontal asymptote y=0 . Their domain is all real numbers and range is positive reals. You can use the xy or ^ button on your calculator to compute values or graph.Tool: Transforming exponential functions. shifts the graph of bx right by h and up by k . If a is negative, the graph flips.Tool: Exponential functions are one-to-one, i.e., if bx=by , then x=y . We can use this property to solve equations. Tool: Compound interest formula is useful for calculating how your investments grow, or how much the credit card companies make off of you, , where P is initial amount of investment, S is amount of investment at time t (in years). r is annual interest rate (APR) and n is number of compounding periods (accrued interest is raked into the pile yearly ( n=1 ), quarterly( n=4 ), monthly( n=12 ), ...). Tool: Continuous compounding formula. If we let n become very large (infinite number of compounding periods) in the compound interest formula, the whole formula reduces to , where e=2.718281828... (there is an ex button on your calculator that will approximate better than memory). NOTE: This same formula is used to model population growth rates, discharge of capacitor, radioactive decay, and lots of other stuff.       3 Properties of Logarithmic Functions Tool: Since exponential functions y=bx are one-to-one, we can solve for input x in terms of output y by invoking inverse operation . Recall with inverse functions the input and output switch places (output becomes input and vice versa). Make sure you identify the base b and don't confuse it with input or output.       Tool: The function is also one-to-one and forms an inverse pair with bx . So  and Tool: Your calculator has two log buttons and . Some logarithms you can evaluate by hand by writing the input as a power of the base.Tool: Common properties of logarithms may be derived from similar properties for exponents   Tool: Change of base formula from base b to base c Normally this is used to convert to log keys on your calculator.           4 Graphing Logarithmic Functions Tool: Graph of is obtained by reflecting y=bx about the line y=x . We can use this symmetry and exponential graphs to generate points on log graphs. Tool: graph may be (h , k ) shifted and reflected via a     5 Exponential and Log Equations/Inequalities Tool: Use one-to-one properties. If bx=by then x=y . Tool: If exponential equation can't be written in terms of powers of the same base, apply log or ln to both sides.         Tool: Quadratic equations are solvable by substituting y=bx and factoring or applying quadratic formula to the resulting equation. Then solve for and check for extraneous solutions.Tool: Use one-to-one property of logarithms. logb x=logb y , then x=y . Tool: Put logarithms together using the properties from section 3 and rewrite in exponential form. Inverses for solving equations: Cube and cube root are inverse pairs. Can you think of any other inverse pairs? What's the inverse of ? Verifying inverses algebraically: Verify that y=2x−2 and are inverses using the definition.   Graphs of inverses: Can you see any relation between graphs of cube and cube root? Can you generate one graph from the other using symmetry? What is inverse graph of the function What are the domains and ranges? Testing inverse graphs via horizontal line test. What is the inverse graph of the function y=x2 + 2 ? Is it a function?   Finding inverse functions (if possible): Find the inverse function of f ( x )=2x−2 Find the inverse function of f ( x )=x2 + 2 Find the inverse function of f ( x )=−3( x−2)3 + 1 Application: The equation for converting degrees Celsius to Fahrenheit is Find the inverse function C (F) for converting from Fahrenheit to Celsius and use it to compute what 90 F is in Celsius. Ans: and C(90) ≈32.22 Exponential graphs. Sketch the graphs of 3x and Do you notice any symmetry? Why do you suppose there might be symmetry? Sketch the transformed exponentials, indicating asymptotes, domain and range.  Solve for x Compound interest problem: Suppose \$100,000 is invested in the bank over 5 years at 5% compounded quarterly. What is the final return? Ans: about \$128,203.72 Suppose you have \$100,000 owed on a credit card over 5 years at 25% compounded quarterly. If by some miracle you don't have late fees, how much will you owe? Ans: about \$ 336,185.34 Continuous compounding: Suppose \$100,000 is invested in the bank over 5 years at 5% compounded continuously. What is the final return? Ans: about \$128,402.54 Capacitor discharge:   describes what happens to voltage V during time t when you charge up a capacitor C with current and then use the energy to warm up a resistor R . For and C=0.001F and initial voltage . What is the voltage after 1 second? After 5 seconds? Ans: 1.839 V after 1 second; 0.037 V after 5 seconds Rearrange from exponential to logarithmic form: Rearrange from logarithmic to exponential form: Evaluating Logarithms  Write each expression as a sum or difference of multiples of logarithms: Write each expression as a single log: Compute: Ans: about -0.6187 Ans: about 3.3030 Application: Acidity is measured by pH which is given by where is hydrogen ion concentration moles per liter. How much more acidic is a substance with pH 3.2 than a substance with pH 5.4? Ans: about 158 times Logarithm graphs: Can you sketch the graph of knowing the graph of 3x? Evaluate points for some small values of x . What happens to the asymptote? Sketch the graph of using the graph of . Then sketch the graph specifying domain, range, and asymptotes.Solve for t : Solve for u : Ans: Solve for x : (a typo on p. 331 says this is not solvable analytically. Actually it is) 9x−3x−12=0 Ans: x=log3 4≈1.2619  Solve for w : Solve for x : Application: Assuming continuous compounding, how long does it double money at 5% interest rate. At 25% interest rate? Ans: about 13.86 years at 5%, about 2.77 years at 25%.