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 f(x)=3x2+5f(x) = 3x - 2 + 5

We start by simplifying this expression: f(x)=3x2+5=3x+3f(x) = 3x - 2 + 5 = 3x + 3

i) Domain and Range of f(x)=3x+3f(x) = 3x + 3

  1. Domain: Since f(x)=3x+3f(x) = 3x + 3 is a linear function, it is defined for all real values of xx. Therefore: Domain of f=(,)\text{Domain of } f = (-\infty, \infty)

  2. Range: Since f(x)=3x+3f(x) = 3x + 3 is also a linear function without any restrictions, it can output any real number as xx varies over all real numbers. Thus: Range of f=(,)\text{Range of } f = (-\infty, \infty)

ii) Relationship Between ff and f1f^{-1}

To find the inverse function f1f^{-1} of f(x)=3x+3f(x) = 3x + 3, follow these steps:

  1. Replace f(x)f(x) with yy: y=3x+3y = 3x + 3

  2. Solve for xx in terms of yy: y3=3xy - 3 = 3x x=y33x = \frac{y - 3}{3}

  3. Replace yy with f1(x)f^{-1}(x): f1(x)=x33f^{-1}(x) = \frac{x - 3}{3}

Now, f(x)=3x+3f(x) = 3x + 3 and f1(x)=x33f^{-1}(x) = \frac{x - 3}{3}.

To represent the relationship between ff and f1f^{-1} graphically, note that their graphs are reflections of each other across the line y=xy = x. Plotting both functions on the same coordinate plane would show this reflective property.


(b) Given Functions g(x)=x+1g(x) = x + 1 and h(x)=ax+3h(x) = ax + 3

For this part, we’ll need additional information about the variable aa or further questions to proceed meaningfully with gg and hh.

Would you like to continue with assumptions about aa or provide additional information for part (b)? Let me know if you’d like a graph as well.


Relative Questions

  1. What is the graphical significance of the line y=xy = x in relation to a function and its inverse?
  2. How do the slopes of f(x)f(x) and f1(x)f^{-1}(x) relate to each other?
  3. Why is the domain and range of a linear function like 3x+33x + 3 unrestricted?
  4. How does the concept of reflection across y=xy = x apply to inverse functions in general?
  5. What changes if f(x)f(x) is a non-linear function in terms of finding its inverse?

Tip

Remember, the inverse function f1f^{-1} exists only if ff is bijective (both injective and surjective). This requirement is met if ff is strictly monotonic.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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