Math Problem Statement

The function f and g are defined by the equations f(x) = 3(x+1)/(2x^2+7x-4) - 1/(x+4) for x > 1/2 and g(x) = ln(x + 1) for x > -1. Solve the following parts: (a) Show that f(x) = 1/(2x - 1). (b) Find f^(-1)(x). (c) Find the domain of f^(-1)(x). (d) Find g^(-1)(x) and its domain. (e) Sketch the curves for y = g(x) and y = g^(-1)(x). (f) Solve f(g(x)) = 1/7.

Solution

Let's break down the steps needed to solve this problem set:

Given:

  • f(x)=3(x+1)2x2+7x41x+4, xR, x>12f(x) = \frac{3(x+1)}{2x^2 + 7x - 4} - \frac{1}{x + 4}, \ x \in \mathbb{R}, \ x > \frac{1}{2}
  • g(x)=ln(x+1), x>1g(x) = \ln(x+1), \ x > -1

(a) Show that f(x)=12x1f(x) = \frac{1}{2x - 1}

This step requires simplifying f(x)f(x).

  1. Step 1: Simplify 3(x+1)2x2+7x4\frac{3(x+1)}{2x^2 + 7x - 4}.

    • Factorize the quadratic denominator if possible.
  2. Step 2: Combine the two terms of f(x)f(x) into one expression.

(b) Find f1(x)f^{-1}(x)

To find the inverse function of f(x)f(x), follow these steps:

  1. Set y=f(x)=12x1y = f(x) = \frac{1}{2x - 1}.
  2. Swap yy and xx to find the inverse.
  3. Solve for yy in terms of xx.

(c) Find the domain of f1(x)f^{-1}(x)

Since the inverse function is f1(x)f^{-1}(x), find the domain by considering the range of the original function f(x)f(x).

(d) Find g1(x)g^{-1}(x) and its domain

Since g(x)=ln(x+1)g(x) = \ln(x + 1), follow these steps to find its inverse:

  1. Set y=g(x)=ln(x+1)y = g(x) = \ln(x + 1).
  2. Solve for xx in terms of yy.

The domain of g1(x)g^{-1}(x) corresponds to the range of g(x)g(x).

(e) Sketch the curves of y=g(x)y = g(x) and y=g1(x)y = g^{-1}(x)

  1. Plot the logarithmic curve y=ln(x+1)y = \ln(x + 1).
  2. Plot the inverse curve.
  3. Identify the points where the curves cross the axes.

(f) Solve f(g(x))=17f(g(x)) = \frac{1}{7}

  1. Use the composite function f(g(x))f(g(x)).
  2. Set f(g(x))=17f(g(x)) = \frac{1}{7} and solve for xx in terms of ee.

Do you want me to show the detailed solution for each part one by one?

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Inverse Functions
Logarithmic Functions
Composite Functions

Formulas

f(x) = 1/(2x - 1)
g(x) = ln(x + 1)
f^(-1)(x) and g^(-1)(x)

Theorems

Inverse Function Theorem
Logarithmic Properties

Suitable Grade Level

Grades 11-12 (Pre-Calculus or Calculus)