Math Problem Statement

f (x) = 5 √x − 3 (c) f (x) = 8x + 1 x2 + 3x + 2 (d) f (x) = 8x + 1 √x2 + 3x + 2 1

Solution

It seems like you have multiple function expressions here that need to be analyzed or simplified. Let’s go through each one individually to clarify and address them properly.

1. Function f(x)=5x3f(x) = 5\sqrt{x} - 3

This function represents a square root function with a linear transformation:

  • The term x\sqrt{x} suggests the function is defined for x0x \geq 0.
  • The transformation 5x5\sqrt{x} scales the output by 5.
  • Subtracting 3 shifts the function downward by 3 units.

2. Function f(x)=8x+1x2+3x+2f(x) = \frac{8x + 1}{x^2 + 3x + 2}

This is a rational function where:

  • The numerator is 8x+18x + 1.
  • The denominator is x2+3x+2x^2 + 3x + 2.
  • The domain is restricted where the denominator is not zero. To find where it’s undefined, set x2+3x+2=0x^2 + 3x + 2 = 0, which factors as (x+1)(x+2)=0(x + 1)(x + 2) = 0. So, x=1x = -1 and x=2x = -2 are not in the domain.

3. Function f(x)=8x+1x2+3x+2f(x) = \frac{8x + 1}{\sqrt{x^2 + 3x + 2}}

This function is similar to the previous one but has a square root in the denominator:

  • The square root restricts the function further since x2+3x+2x^2 + 3x + 2 must be non-negative and cannot be zero.
  • Since the expression under the square root factors as (x+1)(x+2)(x + 1)(x + 2), the domain restrictions are x2x \geq -2 except for x=1x = -1 and x=2x = -2.

Summary of Important Points:

  1. The first function f(x)=5x3f(x) = 5\sqrt{x} - 3 is defined for x0x \geq 0.
  2. The second function f(x)=8x+1x2+3x+2f(x) = \frac{8x + 1}{x^2 + 3x + 2} is undefined at x=1x = -1 and x=2x = -2.
  3. The third function f(x)=8x+1x2+3x+2f(x) = \frac{8x + 1}{\sqrt{x^2 + 3x + 2}} is defined for x2x \geq -2 but not at x=1x = -1 and x=2x = -2.

Would you like me to dive deeper into any of these functions or explore specific properties like asymptotes, intercepts, or graphs?

Related Questions:

  1. How do you determine the domain of a function with a square root?
  2. What are the asymptotes of the function f(x)=8x+1x2+3x+2f(x) = \frac{8x + 1}{x^2 + 3x + 2}?
  3. How do you factor quadratic expressions like x2+3x+2x^2 + 3x + 2?
  4. What is the behavior of f(x)=5x3f(x) = 5\sqrt{x} - 3 as xx approaches infinity?
  5. How do you simplify expressions involving square roots in the denominator?

Tip:

When analyzing the domain of a function, always look out for restrictions such as divisions by zero and square roots of negative numbers.

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Math Problem Analysis

Mathematical Concepts

Algebra
Square Root Functions
Rational Functions
Quadratic Equations

Formulas

f(x) = 5√x − 3
f(x) = (8x + 1) / (x^2 + 3x + 2)
f(x) = (8x + 1) / √(x^2 + 3x + 2)

Theorems

Domain of functions with square roots: x ≥ 0
Domain of rational functions: denominator cannot be zero
Quadratic factorization: x^2 + 3x + 2 = (x + 1)(x + 2)

Suitable Grade Level

Grades 10-12