Functions f and g are defined by f : x ↦ 2 x 2 - 8 x + 10 for 0 ⩽ x ⩽ 2 g : x ↦ x for 0 ⩽ x ⩽ 10 (v) Find an expression for f - 1 ( x ) .

y = 2 x 2 - 8 x + 10 ⇒ y = 2 x 2 - 8 x + 8 - 8 + 10 ⇒ y = 2 ( x 2 - 4 x + 4 ) + 2 ⇒ y - 2 = 2 ( x - 2 ) 2 ⇒ y - 2 2 = ( x - 2 ) 2 ⇒ y - 2 2 = x - 2 ⇒ y - 2 2 + 2 = x Replace x with y ∴ x - 2 2 + 2 = f - 1 ( x )

Functions f and g are defined by f : x ↦ 2 x 2 - 8 x + 10 for 0 ⩽ x ⩽ 2 g : x ↦ x for 0 ⩽ x ⩽ 10 (v) Find an expression for f - 1 ( x ) .

y = 2 x 2 - 8 x + 10 ⇒ y = 2 x 2 - 8 x + 8 - 8 + 10 ⇒ y = 2 ( x 2 - 4 x + 4 ) + 2 ⇒ y - 2 = 2 ( x - 2 ) 2 ⇒ y - 2 2 = ( x - 2 ) 2 ⇒ y - 2 2 = x - 2 ⇒ y - 2 2 + 2 = x Replace x with y ∴ x - 2 2 + 2 = f - 1 ( x )

Draw the graph of a function y = 2 x - 3 for x ∈ ℝ . If the graph represent a function, state whether it is a one-one or many-one function.

Given y = 2 x - 3 for x ∈ ℝ . A function is a relation that uniquely associates members of one set with members of another set. We can see that the input value shown has unique output values. This means that the graph represent a functionAny function f x is defined on set X is said to be one one if the horizontal line cuts the graph of f x at most one point. Hence, the given is one one function.

Draw the graph of a function y = x 2 - 3 for x ∈ ℝ . If the graph represent a function, state whether it is a one-one or many-one function.

Given y = x 2 - 3 for x ∈ ℝ . x x 2 - 3 - 2 1 - 1 - 2 0 - 3 1 - 2 2 1 A function is a relation that uniquely associates members of one set with members of another set. We can see that the input value shown has unique output values. This means that the graph represent a function. The horizontal line test checks if a function is one-to-one . If a horizontal line passes through a graph more than once, the function has more than one x value for at least one y value, so it cannot be one-to-one. A many-one function has one output value for each input value but each output value can have more than one input value. Hence, the given function is many one function.

Draw the graph of a function y = 2 x 3 - 1 for x ∈ ℝ . If the graph represent a function, state whether it is a one-one or many-one function.

Given y = 2 x 3 - 1 for x ∈ ℝ . x 2 x 3 - 1 - 2 - 17 - 1 - 3 0 - 1 1 1 2 15 A function is a relation that uniquely associates members of one set with members of another set. We can see that the input value shown has unique output values. This means that the graph represent a function. The horizontal line test checks if a function is one-to-one . If a horizontal line passes through a graph more than once, the function has more than one x value for at least one y value, so it cannot be one-to-one. Any function f x is defined on set X is said to be one one if the horizontal line cuts the graph of f x at most one point. Hence, the given function is one one function.

Draw the graph of a function y = 2 x for x ∈ ℝ . If the graph represent a function, state whether it is a one-one or many-one function.

Given y = 2 x for x ∈ ℝ . x 2 x - 2 0 . 25 - 1 0 . 5 0 1 1 2 2 4 A function is a relation that uniquely associates members of one set with members of another set. We can see that the input value shown has unique output values. This means that the graph represent a function. The horizontal line test checks if a function is one-to-one . If a horizontal line passes through a graph more than once, the function has more than one x value for at least one y value, so it cannot be one-to-one. Any function f x is defined on set X is said to be one one if the horizontal line cuts the graph of f x at most one point. For example: In the graph, we can observe horizontal line will not intersect more than one point. Hence, the given function is one one function.

Draw the graph of a function y = 10 x for x ∈ ℝ . If the graph represent a function, state whether it is a one-one or many-one function.

Given y = 10 x for x ∈ ℝ . x 10 x - 2 - 5 - 1 - 10 0 undefined 1 10 2 5 A function is a relation that uniquely associates members of one set with members of another set. We can see that the input value shown has unique output values. This means that the graph represent a function. The horizontal line test checks if a function is one-to-one . If a horizontal line passes through a graph more than once, the function has more than one x value for at least one y value, so it cannot be one-to-one. Any function f x is defined on set X is said to be one one if the horizontal line cuts the graph of f x at most one point. For example: In the graph, we can observe horizontal line will not intersect more than one point. Hence, the given function is one one function.

Draw the graph of a function y = 3 x 2 + 4 for x ∈ ℝ , x ≥ 0 . If the graph represent a function, state whether it is a one-one or many-one function.

Given y = 3 x 2 + 4 for x ∈ ℝ , x ≥ 0 . x 3 x 2 + 4 0 4 1 7 2 16 3 31 4 52 Plot the graph using the above table A function is a relation that uniquely associates members of one set with members of another set. We can see that the input value shown has unique output values. This means that the graph represent a function. The horizontal line test checks if a function is one-to-one . If a horizontal line passes through a graph more than once, the function has more than one x value for at least one y value, so it cannot be one-to-one. Any function f x is defined on set X is said to be one one if the horizontal line cuts the graph of f x at most one point. For example: In the graph, we can observe horizontal line will not intersect more than one point. Hence, the given function is one one function.

Draw the graph of a function y = x for x ∈ ℝ , x ≥ 0 . If the graph represent a function, state whether it is a one-one or many-one function.

Given y = x for x ∈ ℝ , x ≥ 0 . x x 0 0 1 1 2 1 . 414 4 2 9 3 Plot the graph using the above table A function is a relation that uniquely associates members of one set with members of another set. We can see that the input value shown has unique output values. This means that the graph represent a function. The horizontal line test checks if a function is one-to-one . If a horizontal line passes through a graph more than once, the function has more than one x value for at least one y value, so it cannot be one-to-one. Any function f x is defined on set X is said to be one one if the horizontal line cuts the graph of f x at most one point. For example: In the graph, we can observe horizontal line will not intersect more than one point. Hence, the given function is one one function.

Draw the graph of a function y 2 = 4 x for x ∈ ℝ . If the graph represent a function, state whether it is a one-one or many-one function.

Given y 2 = 4 x for x ∈ ℝ . x y 0 0 1 2 or - 2 2 2 2 or - 2 2 3 2 3 or - 2 3 4 4 or - 4 Plot the graph using the above table A function is a relation that uniquely associates members of one set with members of another set. We can see that the input value shown has two output value. This means that the graph does not represent a function.