University:
Kentucky State University
Course:
MAT 120 | Precalculus
Academic year:

2024
Views:

451

Pages:

9
Author:

Tyshawn Shaffer

**Exponential Function** **Definition:** f(x) = b^x, b > 0, b ≠ 1 is called an exponential function. **Example:** Sketch f(x) = 2^x | x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | |---|---|---|---|---|---|---|---| | f(x) | 1/8 | 1/4 | 1/2 | 1 | 2 | 4 | 8 | **Domain:** R **Range:** (0, ∞) **Has inverse / one-to-one** **Example:** Sketch f(x) = (1/2)^x | x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | |---|---|---|---|---|---|---|---| | f(x) | 8 | 4 | 2 | 1 | 1/2 | 1/4 | 1/8 | **Domain:** R **Range:** (0, ∞) **Has inverse / one-to-one** **Definition:** The natural exponential function f(x) = e^x where e = 2.718... e^-x = (1/e)^x **Note that:** Dom {e^x} = Dom {f(x)} Dom {e^x} = Dom {f(x)} **Example 7: Find Domains:** 1. f(x) = 3^(x+1) = Dom {x+1} = R 2. f(x) = 2^(1/x-1) = Dom {1/(x-1)} = R - {1} 3. f(x) = e^(√(x-1)) = Dom {√(x-1)} => x-1 ≥ 0 => x ≥ 1 **Note that:** 1. a^x * a^y = a^(x+y) 2. a^x / a^y = a^(x-y) 3. (a^x)^y = a^(xy) 4. a^0 = 1 5. (ab)^x = a^x * b^x 6. (a/b)^x = a^x / b^x 7. a^(1/x) = ∛a **Logarithmic Function** Note that: f(x) = b^x f⁻¹(x) = log_b x **Logarithmic Function:** f(x) = log_b x, b > 0, b ≠ 1 **Domain & Range** log_b x: R⁺ → R, f⁻¹ → R⁺ → R **To find dom log f(x), set f(x) > 0** **Example:** Find dom f(x) = log_3 (x+5)³ x+5 > 0 x > -5 ⇒ (-5, ∞) **f(x) = b^x <--> f⁻¹(x) = log_b x** **f o f⁻¹(x) = x** **f⁻¹ o f(x) = x** **f⁻¹(f(x)) = log_b b^x = x**