Substitute and Simplify: Finding f(5/x) and g(x

Math Problem Statement

Find f(5/x) and g(x - 3) where f(x) = (4x + 3) / (5x + 2) and g(x) = √(x² - 8x).

The image shows two functions $f(x)$ and $g(x)$ , defined as follows:

$f(x) = \frac{4x + 3}{5x + 2}, \quad g(x) = \sqrt{x^2 - 8x}$

You are asked to find:

We substitute $\frac{5}{x}$ into the function $f(x)$ :

= \frac{\frac{20}{x} + 3}{\frac{25}{x} + 2}$$ Now, simplify both the numerator and the denominator: $$= \frac{\frac{20 + 3x}{x}}{\frac{25 + 2x}{x}} = \frac{20 + 3x}{25 + 2x}$$ ### 2. Finding $$g(x - 3)$$: We substitute $$x - 3$$ into the function $$g(x)$$: $$g(x - 3) = \sqrt{(x - 3)^2 - 8(x - 3)}$$ First, expand $$(x - 3)^2$$ and simplify: $$(x - 3)^2 = x^2 - 6x + 9$$ Then: $$g(x - 3) = \sqrt{x^2 - 6x + 9 - 8(x - 3)}$$ $$g(x - 3) = \sqrt{x^2 - 6x + 9 - 8x + 24} = \sqrt{x^2 - 14x + 33}$$ ### Final Answers: 1. $$f\left( \frac{5}{x} \right) = \frac{20 + 3x}{25 + 2x}$$ 2. $$g(x - 3) = \sqrt{x^2 - 14x + 33}$$ Would you like a more detailed explanation on any step? Here are five related questions to deepen your understanding: 1. How do you handle functions with fractions when substituting a complex term? 2. What strategies can you use to simplify square root expressions like $$g(x - 3)$$? 3. How would the function $$f(x)$$ change if the coefficients in the numerator or denominator were different? 4. Can you graph the functions $$f(x)$$ and $$g(x)$$ based on these transformations? 5. How do substitutions in functions affect their domain and range? **Tip:** When simplifying expressions, always check if the result can be further reduced or factored to its simplest form.

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Algebra
Function Substitution
Simplification
Square Roots

f(x) = (4x + 3) / (5x + 2)
g(x) = √(x² - 8x)
f(5/x) = (20 + 3x) / (25 + 2x)
g(x - 3) = √(x² - 14x + 33)

Grades 9-11

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