$$ f\left(x\right)=\frac{5x+3}{6x+7} $$and $$ g\left(x\right)=\sqrt{x^2-4x} $$find $$ f\left(\frac{1}{x}\right) $$and $$ g\left(x-2\right) $$

To find $ f\left(\frac{1}{x}\right) $ and $ g\left(x-2\right) $, we need to substitute the given expressions into the functions $ f(x) $ and $ g(x) $ respectively.

### 1. Finding $ f\left(\frac{1}{x}\right) $:

The function $ f(x) $ is given by:

\[ f(x) = \frac{5x + 3}{6x + 7} \]

To find $ f\left(\frac{1}{x}\right) $, we substitute $ x = \frac{1}{x} $ into the function $ f(x) $:

\[ f\left(\frac{1}{x}\right) = \frac{5\left(\frac{1}{x}\right) + 3}{6\left(\frac{1}{x}\right) + 7} \]

Now, simplify the expression:

\[ f\left(\frac{1}{x}\right) = \frac{\frac{5}{x} + 3}{\frac{6}{x} + 7} \]

Multiply the numerator and the denominator by $ x $ to clear the fractions:

\[ f\left(\frac{1}{x}\right) = \frac{5 + 3x}{6 + 7x} \]

### 2. Finding $ g\left(x-2\right) $:

The function $ g(x) $ is given by:

\[ g(x) = \sqrt{x^2 - 4x} \]

To find $ g\left(x-2\right) $, we substitute $ x = x-2 $ into the function $ g(x) $:

\[ g\left(x-2\right) = \sqrt{\left(x-2\right)^2 - 4\left(x-2\right)} \]

First, expand $ \left(x-2\right)^2 $:

\[ \left(x-2\right)^2 = x^2 - 4x + 4 \]

Then, distribute the $ -4 $:

\[ -4\left(x-2\right) = -4x + 8 \]

Combine these results:

\[ g\left(x-2\right) = \sqrt{x^2 - 4x + 4 - 4x + 8} \]

Simplify inside the square root:

\[ g\left(x-2\right) = \sqrt{x^2 - 8x + 12} \]

### Summary of Results:

- $ f\left(\frac{1}{x}\right) = \frac{5 + 3x}{6 + 7x} $
- $ g\left(x-2\right) = \sqrt{x^2 - 8x + 12} $

Would you like details or have any questions? 

Here are 8 related questions to expand your understanding:

1. What are the domain restrictions for the function $ f(x) = \frac{5x + 3}{6x + 7} $?
2. What are the domain restrictions for $ f\left(\frac{1}{x}\right) $?
3. How do you find the domain of the function $ g(x) = \sqrt{x^2 - 4x} $?
4. What are the domain restrictions for $ g\left(x-2\right) $?
5. How does transforming $ x $ to $ \frac{1}{x} $ affect the graph of a function?
6. How does shifting $ x $ by 2 units affect the graph of a function?
7. How do you simplify the expression $ \frac{a/x + b}{c/x + d} $?
8. What is the effect of shifting the input of a square root function by a constant value?

**Tip:** When dealing with transformations of functions, always pay close attention to how substitutions affect both the algebraic form and the domain of the function.

Explore how to compute functions f(1/x) and g(x-2) step-by-step with detailed solutions. Understand the impact of substitution on function values and the effect of shifting inputs on square root functions. Suitable for Grades 10-12.

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