Exponential functions
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The document discusses exponential functions and exponential equations. Exponential functions have the form f(x) = bx, where b is the base and x is the exponent. These functions are important in modeling real-world phenomena like population growth. Exponential equations set the exponents of the same base equal to solve for the variable. They can be solved by rewriting all terms to have the same base, setting the exponents equal, and solving the resulting equation.
Exponential functions
- 2. The exponential function is very important in math because it is used to model many real life situations. ◦ For example: population growth and decay, compound interest, economics, and much more.
- 3. f ( x) b x Question… How is this function different from functions that we have worked with previously?
- 4. Answer: The exponent is the variable! f ( x) b x Question… b = The base b >0 and b ≠ 1 x = The exponent x=any real number Is f(x)=x3 an exponential function? NO
- 5. One of the most common exponential functions is x f ( x) 2 The graph looks like this:
- 6. f ( x) 2 x The graph starts off slow but then increases very rapidly The x-axis (y=0) is an asymptote. X can be any real number, for example: f (x) 2 3 (0,1) is the y intercept Models Exponential Growth
- 7. Question… What would a graph look like if b is between 0 and 1? For example: x f ( x) 1 2
- 8. f ( x) 1 2 x The graph starts off very high but then decreases very rapidly The x-axis (y=0) is an asymptote. X can be any real number, for example: f (x) 1 2 (0,1) is the y intercept Models Exponential Decay
- 9. Definition f ( x) b b>1 x b = The base b >0 and b ≠ 1 x = The exponent x=any real number 0<b<1
- 11. What is an exponential equation? ◦ An equation where the variable is the exponent. ◦ Example: 2 x 1 8
- 12. 2 1 8 x Any ideas? What if we changed the right side to 2 x 2 3 2 3 Now What?
- 13. x 2 ◦If 2 ◦Then x=? ◦Check: -3= 1 ◦2 8 3 3 Then we have solved 2 x 1 8
- 14. 1.Express each side of the equation as a power of the same base. 2.Set the exponents equal and solve. 3.Check your answer Main idea If the bases are the same, set the exponents equal!
- 15. Solve for x: 7 x 2 7 3x Are the bases equal? YES Set the exponents equal. Solve for x. x 2 3x All we have here is a simple Algebra problem x 1
- 16. x = -1 7 7 x 2 ( 1) 2 7 3 3x 7 3( 1) 7 3 7 It checks!
- 17. Solve for x: 5 4 t t 1 25 Are the bases equal? NO Change the right side to: Simplify: Solve! 5 4 t t 5 2 2t 2 2 t 1 (5 )
- 18. t=2 4 t t 1 5 25 4 2 2 1 5 25 2 1 5 25 It checks!
- 19. Solve for x: 49 x 2 7 7 11 x 4
- 20. Solve for x: 49 x 2 7 7 Are the bases equal? NO 2 x Change both sides to: (7 ) Simplify: Solve! 7 2x 4 2 7*7 3 2 7 3 2x 4 2 11 x 4 1 2
- 21. x = 11/4 x 2 49 11 49 4 2 3 4 3 2 4 73 7 7 23 2 3 2 49 7 (7 ) 7 It checks!
- 22. REVIEW 1. Exponential function: f ( x) b Key Point: 2. Exponential Equations: Example: Rewrite as: x The exponent is the variable b= the base b >0 and b ≠1 X= the exponent X = any real number An equation where the exponent is the variable 4x 6 2 16 4x 6 4 2How to solve: 2 Set exponents 4x 6 4 If the bases are the equal: Check: same, set the10 exponents x Solve: equal! 4 2 4( 10 ) 6 4 10 6 2 2 4 16 16 16 It checks!