Problem 24 - A Graphical Approach to Precalculus with Limits

Question

For each pair of fimetions, (a) find ( \(f+g)(x),(f-g)(x),\) and \((f g)(x) ;\) (b) give the domains of the functions in part (a); (c) find \(\frac{t}{x}\) and give its domain; (d) find \(f \circ g\) and give is domain: and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=9-2 x, g(x)=-5 x+2$$

Step-by-Step Solution

Answer

Functions are calculated and domains found. The domains are \( \mathbb{R} \) except for \( \frac{f}{g}(x) \) which excludes \( x = \frac{2}{5} \).

Step 1: Calculate (f + g)(x)

To find \((f + g)(x)\), add the expressions for \(f(x)\) and \(g(x)\) together: \[f(x) + g(x) = (9 - 2x) + (-5x + 2)\]. Simplify the expression by combining like terms: \[f(x) + g(x) = 9 + 2 - 2x - 5x = 11 - 7x\]. So, \((f + g)(x) = 11 - 7x\).

Step 2: Calculate (f - g)(x)

To find \((f - g)(x)\), subtract the expression for \(g(x)\) from that for \(f(x)\): \[(9 - 2x) - (-5x + 2)\]. Simplify by distributing the negative sign: \[9 - 2x + 5x - 2 = 7 + 3x\]. So, \((f - g)(x) = 7 + 3x\).

Step 3: Calculate (f * g)(x)

To find \((f \cdot g)(x)\), multiply the expressions for \(f(x)\) and \(g(x)\): \[(9 - 2x)(-5x + 2)\]. Apply the distributive property: \[9(-5x) + 9\cdot2 + (-2x)(-5x) + (-2x)2 = -45x + 18 + 10x^2 - 4x\]. Combine like terms: \[10x^2 - 49x + 18\]. So, \((f \cdot g)(x) = 10x^2 - 49x + 18\).

Step 4: Determine Domains for (f+g), (f-g), (f * g)

The domains of \((f+g)(x)\), \((f-g)(x)\), and \((f \cdot g)(x)\) are all real numbers \( \mathbb{R} \) because there are no restrictions such as division by zero or square roots of negative numbers in any of these expressions.

Step 5: Calculate f/g(x)

To find \( \frac{f}{g}(x) \), use \[ \frac{9-2x}{-5x+2} \]. This expression is defined as long as the denominator \(-5x+2 eq 0\). So, \(-5x+2 = 0\implies -5x = -2 \implies x = \frac{2}{5}\). The function's domain is all real numbers except \( x = \frac{2}{5} \).

Step 6: Calculate f ∘ g(x)

To find \( f \circ g(x) \), substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(-5x + 2) = 9 - 2(-5x + 2)\]. Simplify: \[9 + 10x - 4 = 10x + 5\]. The domain is all real numbers \( \mathbb{R} \), as no restrictions exist in the expression.

Step 7: Calculate g ∘ f(x)

To find \( g \circ f(x) \), substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(9 - 2x) = -5(9 - 2x) + 2\]. Simplify: \[-45 + 10x + 2 = 10x - 43\]. The domain is all real numbers \( \mathbb{R} \), as no restrictions exist in the expression.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when dealing with mathematical expressions. The domain of a function refers to all the possible input values (Such as numbers you can plug into a function without causing any issues like division by zero or square roots of negative numbers. For functions like polynomials, which are expressed in standard algebraic form, the domain usually includes all real numbers. However, when dealing with functions involving fractions or radicals, the domain needs to be restricted to ensure outputs are real and defined.

• For example, in the function \(\frac{f}{g}(x) = \frac{9-2x}{-5x+2}\), the domain excludes the value where the denominator becomes zero. This would be when \(-5x + 2 = 0\), solving for \(x\) gives \(x = \frac{2}{5}\).
• For composite functions like \(f(g(x))\) or \(g(f(x))\), the domain is determined based on the domain of the inside function and any restrictions imposed by the outside function.

The domain defines the scope of the function, ensuring you only use valid inputs to derive consistent outputs.

Composite Functions

Composite functions combine two functions into one by applying one function to the result of another. This may sound complex, but it's just like connecting two machines where the output of the first becomes the input of the second. The notation \(f \circ g(x)\) means "apply \(g(x)\)" first, then use the result as the input for \(f(x)\). To find \(f \circ g(x)\) with \(f(x)=9-2x\) and \(g(x)=-5x+2\), you substitute \(g(x)\) into \(f\):

• First, calculate \(g(x) = -5x + 2\).
• Then, substitute this into \(f(x)\): \(f(-5x + 2) = 9 - 2(-5x + 2) = 10x + 5\).

Composite functions have domains that are the intersection of the domains of \(f\) and \(g\) with additional considerations if \(f\) has restrictions. Understanding composite functions involves ensuring the operations only include values that are valid for both individual functions and the combined operation.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, built using addition, subtraction, multiplication, and non-negative integer exponents. They are the backbone of algebra and appear as foundational elements in many mathematical concepts. They come in various degrees, depending on the highest power of the variable present.

• A first-degree polynomial is a linear function. For example, \(f(x) = 9 - 2x\) is a first-degree polynomial.
• A second-degree polynomial is a quadratic function. An example from the given exercise is the product \((f \cdot g)(x) = 10x^2 - 49x + 18\).

The domain of a polynomial is typically all real numbers because there are no operations that would make the function undefined, such as division by zero. Polynomials are often used in equations to model real-world problems due to their predictable, continuous nature, and they form the basis for higher-level mathematics, including calculus.