The functions f and g are defined such that and g.
Show that
Given that find the value of a.
Show that
Given that find the value of b.
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The functions f and g are defined such that and g.
Show that
Given that find the value of a.
Show that
Given that find the value of b.
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The functions f(x) and g(x) are defined as follows
Write down the range of f(x) .
Find
(i)
Solve the equation f(x) = g(x).
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The graph of y = f(x) is shown below.
On the diagram above sketch the graph of y = f −1(x).
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The function is defined as
≠ 0
Show that can be written in the form
Explain why the inverse of does not exist and suggest an adaption to its domain so the inverse does exist.
The domain of is changed to .
Find an expression for and state its domain and range.
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The functions and are defined as follows
Find
(i)
Write down and state its domain and range.
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A function is defined by
Find the value of
Write down the range of .
Find the inverse function
Write down the range of the inverse function.
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Consider the function The domain of is
Find
Find the range of .
Write down the domain of the inverse function.
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Let , for ≠ 3.
For the graph of , find:
Find the value of
Given that g, find the domain and range of g.
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The functions f and g are defined for by and , where .
Find the range of f.
Given that is always positive for all x, determine the set of possible values for d.
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Let , where ≠ , .
Write down
For the graph of , find the equations of all the asymptotes.
Find
For the graph of , find the equation of
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Let for
Write down an expression for the inverse function .
Consider another function g, for where k is an integer to be found.
Given that the graph of g has an inverse, find the value of .
Sketch the graphs of and g, for the domain found in part (b), on the same set of axes, along with their inverses.
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Consider the function defined by .
Sketch the graph of . Clearly label the points where the graph intersects the axes, along with any points that are local maxima or minima.
Let the function g be defined by g
Given that g has an inverse:
Let the function be defined by
Given that has an inverse:
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A function is called a self-inverse function if for all values of in the domain.
Let , where ≠ 0, .
Show that is a self-inverse function.
Let g, where ≠ ,.
Find the value of .
Show that g is a self-inverse function.
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The functions f and g are defined such that and .
Find , giving your answer in the form where m, h and k are constants to be found.
Hence, or otherwise, find the coordinates of the vertex of the graph of .
Find , giving your answer in the form where and are constants to be found.
Hence, or otherwise, find the coordinates of the y-intercept of the graph of
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Let and , where each function has the largest possible valid domain.
Write down the range of f.
Write down the domain and range of g.
Find
Solve the equation
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The function f is defined by , for .
Write down the range of f.
Write down an expression for
Write down the domain and range of.
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The perimeter, P, and area, A, of a given square can be expressed by and respectively, where x is the length of the side of the square.
Write down an expression for:
Find the value of k and .
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The values of two functions, f and g, for certain values of x are given in the following table:
x |
-2 |
0 |
3 |
f(x) |
-12 |
-4 |
8 |
g(x) |
0 |
-12 |
30 |
Find the value of f -18.
Find the value of
Given that f (x) is a linear function, find f (x).
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Let , for .
Find f -1(2).
Let g be a function such that g-1 exists for all real numbers.
Given that g(14) = 3, find .
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Let the function f be defined by, where f has its largest possible valid domain.
Find the domain and range of f.
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Let and, both for .
Find
Find in the form .
Solve the equation
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Given that g and , find a possible expression for .
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The function g defined by g has an inverse.
Solve = 21.
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The function is defined by .
It is always true that the graphs of a function and its inverse will be reflections of each other in the line .
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The functions f and g are defined such that and , both for .
Find, giving your answer in the form .
Hence, or otherwise, find the x-intercepts of the graph of .
Let .
Find the distance between the y-intercept of the graph of and the positive x-intercept of the graph of . Your answer should be given as an exact value.
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Let the function f be such that .
It is given that the inverse function f -1 exists, and that the domain of f is as large as possible,
suggest two possible domains for f and write down the corresponding ranges.
Find what the value of would be for each of the domains suggested in part (a).
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Let.
Write down the coordinates of the y-intercept of the graph of .
Given that f has the largest possible valid domain,
find the domain and range of f .
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Let the function f be defined by , where k is a constant and where f has the largest possible valid domain.
Find the domain of f.
Given that as gets large tends towards the value −7, find the value of .
Write down the equations of any vertical and/or horizontal asymptotes on the graph of .
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The following diagram shows the graph of , for a function f that has the domain. Point A has coordinates (-3, 2.5) and point B has coordinates (3,-2.5). The x-intercept of the function is (2, 0) as shown.
f can be written as a piecewise function, where each of the two pieces is a linear function and where the domain of the first function is .
Write down as a piecewise function.
Sketch the graph of on the same grid above.
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Consider the function defined by .
Given that and that , find a possible expression for g.
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The functions f and g are defined such that and , both for .
Giving your answers in the form , find
Describe a single transformation that would map the graph of onto the graph of .
Given that , find the value of p.
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Let the functions f and g be defined by and , both for .
Find
Find in the form .
Solve the equation
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A rectangle has length and width .
Find an expression for
Show that .
The graph of the function P, for , is shown below.
On the grid above, draw the graph of the inverse function .
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Consider the function defined by , where is the largest value such that has an inverse.
Find the inverse function .
Let the function g be defined by g.
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A part of the graph of the function is shown below.
Explain why does not have an inverse.
The domain of is now restricted to where and . and are chosen so that has an inverse and the interval is as large as possible.
Find the domain and range of
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