Elementary Functions Part 1, Functions Lecture 1.6d, Function Inverses: One-to-one and onto functions Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 26 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output. Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C: Smith (SHSU) Elementary Functions 2013 28 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output. Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C: Smith (SHSU) Elementary Functions 2013 28 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output. Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C: Smith (SHSU) Elementary Functions 2013 28 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output. Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C: Smith (SHSU) Elementary Functions 2013 28 / 33 Function Inverses We can define one-to-one function more formally, as a mathematical implication: A function f is one-to-one if f(a) = f(b) =) a = b: One can think of this implication this way: \A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same." Let's show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps: 1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5: 3 Subtract 5 from both sides to get 3a = 3b: 4 Divide both sides by 3 to get a = b: Since we have shown that f(a) = f(b) =) a = b, the function f(x) is one-to-one. Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses We can define one-to-one function more formally, as a mathematical implication: A function f is one-to-one if f(a) = f(b) =) a = b: One can think of this implication this way: \A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same." Let's show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps: 1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5: 3 Subtract 5 from both sides to get 3a = 3b: 4 Divide both sides by 3 to get a = b: Since we have shown that f(a) = f(b) =) a = b, the function f(x) is one-to-one. Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses We can define one-to-one function more formally, as a mathematical implication: A function f is one-to-one if f(a) = f(b) =) a = b: One can think of this implication this way: \A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same." Let's show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps: 1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5: 3 Subtract 5 from both sides to get 3a = 3b: 4 Divide both sides by 3 to get a = b: Since we have shown that f(a) = f(b) =) a = b, the function f(x) is one-to-one. Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses We can define one-to-one function more formally, as a mathematical implication: A function f is one-to-one if f(a) = f(b) =) a = b: One can think of this implication this way: \A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same." Let's show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps: 1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5: 3 Subtract 5 from both sides to get 3a = 3b: 4 Divide both sides by 3 to get a = b: Since we have shown that f(a) = f(b) =) a = b, the function f(x) is one-to-one. Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses We can define one-to-one function more formally, as a mathematical implication: A function f is one-to-one if f(a) = f(b) =) a = b: One can think of this implication this way: \A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same." Let's show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps: 1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5: 3 Subtract 5 from both sides to get 3a = 3b: 4 Divide both sides by 3 to get a = b: Since we have shown that f(a) = f(b) =) a = b, the function f(x) is one-to-one. Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses We can define one-to-one function more formally, as a mathematical implication: A function f is one-to-one if f(a) = f(b) =) a = b: One can think of this implication this way: \A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same." Let's show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps: 1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5: 3 Subtract 5 from both sides to get 3a = 3b: 4 Divide both sides by 3 to get a = b: Since we have shown that f(a) = f(b) =) a = b, the function f(x) is one-to-one. Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses We can define one-to-one function more formally, as a mathematical implication: A function f is one-to-one if f(a) = f(b) =) a = b: One can think of this implication this way: \A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same." Let's show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps: 1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5: 3 Subtract 5 from both sides to get 3a = 3b: 4 Divide both sides by 3 to get a = b: Since we have shown that f(a) = f(b) =) a = b, the function f(x) is one-to-one. Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses We can define one-to-one function more formally, as a mathematical implication: A function f is one-to-one if f(a) = f(b) =) a = b: One can think of this implication this way: \A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same." Let's show that the function f(x) = 3x + 5 defined earlier is one-to-one.