Math Problem Statement

Solve the following mathematical problems from the provided image.

Solution

I can help you solve or explain any of the questions from the image you provided. Below is a summary of the key problems from the paper:

  1. Solve the following equations using suitable methods: a. 2xx2x2=13x23\frac{2}{x} - \frac{x - 2}{x^2} = \frac{1}{3x} - \frac{2}{3} b. 11+1x=111x\frac{1}{1 + \frac{1}{x}} = \frac{1}{1 - \frac{1}{x}} c. log2+log(x+2)log(3x5)=log3\log{2} + \log(x + 2) - \log(3x - 5) = \log{3} d. 2x2x2<02x^2 - x - 2 < 0

  2. Find the domain and range of the following functions: a. f(x)=1+8x2x2f(x) = 1 + 8x - 2x^2 b. f(x)=1x25x+6f(x) = \frac{1}{x^2 - 5x + 6} c. f(x)=x26x+8f(x) = \sqrt{x^2 - 6x + 8}

  3. Show whether the following functions are one-to-one, onto, or neither: a. f:RR,f(x)=xf: \mathbb{R} \to \mathbb{R}, \, f(x) = |x| b. f:RR,f(x)=x2f: \mathbb{R} \to \mathbb{R}, \, f(x) = x^2

  4. Find bb and cc given that the graph of y=x2+bx+cy = x^2 + bx + c passes through the points (3, 14) and (-4, 7).

  5. Show that 3+53 + \sqrt{5} is irrational and simplify x(logylogz)x (\log_y{\log z}).

  6. Let f(x)=14x23f(x) = 14x - 23, g(x)=1x2g(x) = \frac{1}{x^2}, h(x)=x2xh(x) = x^2 - x. Find f(g(x)),g(2),h(h(10))f(g(x)), g(2), h(h(10)).

  7. Draw the graph of the following functions: a. f(x)=x4x34x2+x+1f(x) = x^4 - x^3 - 4x^2 + x + 1 b. f(x)=x3x2f(x) = \frac{x}{3x^2} c. f(x)=x2(x1)(x+1)f(x) = \frac{x - 2}{(x - 1)(x + 1)}

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Math Problem Analysis

Mathematical Concepts

Algebra
Logarithms
Quadratic Equations
Inequalities
Function Properties
Irrational Numbers
Graphing

Formulas

Logarithmic identities
Quadratic formula
Inequalities solution methods
Domain and range of functions
One-to-one and onto function properties
Simplification of irrational numbers

Theorems

Properties of logarithms
Quadratic formula theorem
One-to-one and onto function theorem
Irrational numbers theorem

Suitable Grade Level

Grades 10-12