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## 1.6 – Inverse Functions and Logarithms

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**One-To-One Functions**A function is one-to-one if no two domain values correspond to the same range value. Algebraically, a function is one-to-one if f (x1) ≠ f (x2) for all x1 ≠ x2. Graphically, a function is one-to-one if its graph passes the horizontal line test. That is, if any horizontal line drawn through the graph of a function crosses more than once, not it is one-to-one.**Try This**Determine if the following functions are one-to-one. (a) f (x) = 1 + 3x – 2x 4 (b) g(x) = cos x + 3x 2 (c) (d)**Inverse Functions**The inverse of a one-to-one function is obtained by exchanging the domain and range of the function. The inverse of a one-to-one function f is denoted with f -1. Domain of f = Range of f -1 Range of f = Domain of f-1 f −1(x) = y<=> f (y) = x**Try This**Sketch a graph of f (x) = 2x and sketch a graph of its inverse. What is the domain and range of the inverse of f. Domain: (0, ∞) Range: (-∞, ∞)**Inverse Functions**You can obtain the graph of the inverse of a one-to-one function by reflecting the graph of the original function through the line y = x.**Inverse of a One-To-One Function**• To obtain the formula for the inverse of a function, do the following: • Let f (x) = y. • Exchange y and x. • Solve for y. • Let y = f −1(x).**Inverse Functions**Determine the formula for the inverse of the following one-to-one functions. (a) (b) (c)**Logarithmic Functions**The inverse of an exponential function is called a logarithmic function. Definition: x = a y if and only if y = log ax**Logarithmic Funcitons**The function f (x) = log ax is called a logarithmic function. Domain: (0, ∞) Range: (-∞, ∞) Asymptote:x = 0 Increasing for a > 1 Decreasing for 0 < a < 1 Common Point: (1, 0)**Logarithmic Functions**Now determine the inverse of g(x) = 3x. Definition: x = a y if and only if y = log ax**Properties of Logarithms**• log a (ax) = x for all x • alog a x = x for all x > 0 • log a (xy) = log ax + log a y • log a(x / y) = log ax – log a y • log axn = nlog ax • loga ax = x Common Logarithm: log 10x = log x Natural Logarithm: log ex = ln x All the above properties hold.**Properties of Logarithms**The natural and common logarithms can be found on your calculator. Logarithms of other bases are not. You need the change of base formula. where b is any other appropriate base.**Try These**• Determine the exact value of log 8 2. • Determine the exact value of ln e2.3. • Evaluate log 7.3 5 to four decimal places. • Write as a single logarithm: ln x + 2ln y – 3ln z. • Solve 2x +5 = 3 for x.**Try This**In the theory of relativity, the mass of a particle with velocity v is where m0 is the mass of the particle and c is the speed of light in a vacuum. Find the inverse function of f and explain its meaning.