Math Problem Statement
Given that :x In (3x-2)+5,
i) Find, and state its domain and range.
ii) Hence, show the relationship between the graphs of f and f¹ on a diagram.
(b) The function g and h are defined by g(x)=x+1 and h(x)=ax+3, Given
Solution
To tackle this problem, let's go through each part systematically. Based on your description, I'll make some assumptions about missing details and provide solutions accordingly.
Given Function
We start by simplifying this expression:
i) Domain and Range of
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Domain: Since is a linear function, it is defined for all real values of . Therefore:
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Range: Since is also a linear function without any restrictions, it can output any real number as varies over all real numbers. Thus:
ii) Relationship Between and
To find the inverse function of , follow these steps:
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Replace with :
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Solve for in terms of :
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Replace with :
Now, and .
To represent the relationship between and graphically, note that their graphs are reflections of each other across the line . Plotting both functions on the same coordinate plane would show this reflective property.
(b) Given Functions and
For this part, we’ll need additional information about the variable or further questions to proceed meaningfully with and .
Would you like to continue with assumptions about or provide additional information for part (b)? Let me know if you’d like a graph as well.
Relative Questions
- What is the graphical significance of the line in relation to a function and its inverse?
- How do the slopes of and relate to each other?
- Why is the domain and range of a linear function like unrestricted?
- How does the concept of reflection across apply to inverse functions in general?
- What changes if is a non-linear function in terms of finding its inverse?
Tip
Remember, the inverse function exists only if is bijective (both injective and surjective). This requirement is met if is strictly monotonic.
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Math Problem Analysis
Mathematical Concepts
Functions
Linear Functions
Inverse Functions
Domain and Range
Formulas
f(x) = 3x + 3
f⁻¹(x) = (x - 3) / 3
Theorems
Properties of linear functions
Inverse function properties and reflection across y = x
Suitable Grade Level
Grades 10-12