Question

Determine \(h(x)=f(x) g(x)\) and \(k(x)=\frac{f(x)}{g(x)}\) for each pair of functions. a) \(f(x)=x+7\) and \(g(x)=x-7\) b) \(f(x)=2 x-1\) and \(g(x)=3 x+4\) \(f(x)=\sqrt{x+5}\) and \(g(x)=x+2\) d) \(f(x)=\sqrt{x-1}\) and \(g(x)=\sqrt{6-x}\)

Short answer

a) \(h(x)=x^2-49\), \(k(x)=\frac{x+7}{x-7}\)b) \(h(x)=6x^2+5x-4\), \(k(x)=\frac{2x-1}{3x+4}\)c) \(h(x)=\sqrt{x+5}(x+2)\), \(k(x)=\frac{\sqrt{x+5}}{x+2}\)d) \(h(x)=\sqrt{(x-1)(6-x)}\), \(k(x)=\sqrt{\frac{x-1}{6-x}}\)

Step-by-step solution

Define the product function

For each pair of functions, determine the product function, \[h(x) = f(x)g(x)\]. This means you will multiply the expressions for each given pair.

Define the quotient function

Now, determine the quotient function, \[k(x) = \frac{f(x)}{g(x)}\]. This means you will divide the expressions for each given pair.

Solve for pair (a)

Given \(f(x) = x + 7\) and \(g(x) = x - 7\), the product will be:\[h(x) = (x + 7)(x - 7) = x^2 - 49\]The quotient will be:\[k(x) = \frac{x+7}{x-7}\]

Solve for pair (b)

Given \(f(x) = 2x - 1\) and \(g(x) = 3x + 4\), the product will be:\[h(x) = (2x - 1)(3x + 4) = 6x^2 + 8x - 3x - 4 = 6x^2 + 5x - 4\]The quotient will be:\[k(x) = \frac{2x-1}{3x+4}\]

Solve for pair (c)

Given \(f(x) = \sqrt{x + 5}\) and \(g(x) = x + 2\), the product will be:\[h(x) = \sqrt{x+5}(x+2)\]The quotient will be:\[k(x) = \frac{\sqrt{x+5}}{x+2}\]

Solve for pair (d)

Given \(f(x) = \sqrt{x - 1}\) and \(g(x) = \sqrt{6 - x}\), the product will be:\[h(x) = \sqrt{x - 1} \cdot \sqrt{6 - x} = \sqrt{(x-1)(6-x)}\]The quotient will be:\[k(x) = \frac{\sqrt{x-1}}{\sqrt{6-x}} = \sqrt{\frac{x-1}{6-x}}\]

Key concepts

Product of Functions

In precalculus, the product of two functions involves multiplying their expressions together. This is expressed as \( h(x) = f(x)g(x) \). For example, if you have functions \( f(x) = x+7 \) and \( g(x) = x-7 \), the product is \( h(x) = (x+7)(x-7) = x^2 - 49 \). The process is straightforward. You simply multiply everything in the first function by everything in the second function. Remember to combine like terms and simplify the result.

• Step-by-step simplification
• Multiplying terms
• Combining like terms

This operation is used in various applications, such as finding the intersection of two functions when graphed, among others.

Quotient of Functions

Similarly, the quotient of two functions is found by dividing one function by the other, denoted as \( k(x) = \frac{f(x)}{g(x)} \). For example, given \( f(x) = x+7 \) and \( g(x)=x-7 \), you get the quotient \( k(x) = \frac{x+7}{x-7} \). Be mindful of the domain restrictions here because you cannot divide by zero. So for \( g(x) = 0 \), ensure any values that make \( g(x) \) zero are excluded from the domain of the quotient function.

• Performing division
• Checking for undefined values
• Applying domain restrictions

This operation is vital for understanding function behavior and solving rational expressions in higher mathematics.

Composite Functions

Composite functions take one function and apply it within another. This is denoted as \( (f \circ g)(x) \) or simply \( f(g(x)) \). For example, if \( f(x) = 2x-1 \) and \( g(x) = 3x+4 \), the composite function \( f(g(x)) \) becomes \( f(3x+4) = 2(3x+4) - 1 = 6x + 8 - 1 = 6x + 7 \). You substitute the entire \( g(x) \) expression into \( f(x) \).

• Substituting one function into another
• Simplifying the resulting expression
• Understanding function application

Being adept at this enhances your ability to solve more complex functions and is essential in calculus for chain rule and function transformations.

Precalculus Problem-Solving

In precalculus, problem-solving often involves combining multiple concepts like the product, quotient, and composition of functions. Understanding these methods allows students to tackle real-world applications and higher-level math problems efficiently.

• Identifying which operations to use
• Simplifying expressions correctly
• Applying these skills to solve complex equations

Effective problem-solving strategies are crucial for success in mathematics, as well as in various scientific and engineering fields where these principles are applied daily.