Math Problem Statement

If f(x) = sqrt(x^2 + 4), find: a) f(-3), b) f(2), c) f(x + b * x)

Solution

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The problem given is:

If f(x)=x2+4f(x) = \sqrt{x^2 + 4}, Find:
a) f(3)f(-3)
b) f(2)f(2)
c) f(x+bx)f(x + b \cdot x)

Let's go through each part:

a) f(3)f(-3)

Substitute x=3x = -3 into the function: f(3)=(3)2+4=9+4=13f(-3) = \sqrt{(-3)^2 + 4} = \sqrt{9 + 4} = \sqrt{13} So, f(3)=13f(-3) = \sqrt{13}.

b) f(2)f(2)

Substitute x=2x = 2 into the function: f(2)=(2)2+4=4+4=8=22f(2) = \sqrt{(2)^2 + 4} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} So, f(2)=22f(2) = 2\sqrt{2}.

c) f(x+bx)f(x + b \cdot x)

Here, substitute x+bxx + b \cdot x into the function f(x)f(x): f(x+bx)=(x+bx)2+4f(x + b \cdot x) = \sqrt{(x + bx)^2 + 4} Simplify the expression inside the square root: (x+bx)2=x2(1+b)2(x + bx)^2 = x^2(1 + b)^2 Thus, the function becomes: f(x+bx)=x2(1+b)2+4f(x + b \cdot x) = \sqrt{x^2(1 + b)^2 + 4}

Would you like further clarification on any part?

Here are five related questions for further practice:

  1. How would the expression change if the function was f(x)=x2+4f(x) = x^2 + 4 instead of f(x)=x2+4f(x) = \sqrt{x^2 + 4}?
  2. What is f(0)f(0) for the given function f(x)=x2+4f(x) = \sqrt{x^2 + 4}?
  3. How does the function behave for very large values of xx?
  4. What would the derivative f(x)f'(x) be for f(x)=x2+4f(x) = \sqrt{x^2 + 4}?
  5. How would the function change if the +4+4 term were replaced by 4-4?

Tip: When working with square root functions, ensure the expression under the root is non-negative to maintain real-valued outputs.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Square Roots

Formulas

f(x) = sqrt(x^2 + 4)
Substitution: f(x) = sqrt((-3)^2 + 4), f(x) = sqrt(2^2 + 4), f(x) = sqrt((x + bx)^2 + 4)

Theorems

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Suitable Grade Level

Grades 9-11