In Exercises 31-34, find formulas for f+g . f-g . f g, and f...

In Exercises $31-34,$ find formulas for $f+g . f-g . f g,$ and $f / g,$ and state the domains of the functions. $$f(x)=2 x+1, g(x)=x^{2}-x$$ Let $g(x)=\sqrt{x} .$ Find (a) $g(5 s+2)$ (b) $g(\sqrt{x}+2)$ (c) $3 g(5 x)$ (d) $\frac{1}{g(x)}$ (e) $g(g(x))$ (f) $(g(x))^{2}-g\left(x^{2}\right)$ (g) $g(1 / \sqrt{x})$ (h) $g\left((x-1)^{2}\right)$

In Exercises $31-34,$ find formulas for $f+g . f-g . f g,$ and $f / g,$ and state the domains of the functions.
$$f(x)=2 x+1, g(x)=x^{2}-x$$
Let $g(x)=\sqrt{x} .$ Find
(a) $g(5 s+2)$
(b) $g(\sqrt{x}+2)$
(c) $3 g(5 x)$
(d) $\frac{1}{g(x)}$
(e) $g(g(x))$
(f) $(g(x))^{2}-g\left(x^{2}\right)$
(g) $g(1 / \sqrt{x})$
(h) $g\left((x-1)^{2}\right)$

Key Concepts

Function Operations

Function operations involve combining two or more functions using arithmetic operations such as addition, subtraction, multiplication, and division. This process creates new functions where the output values are computed by performing the corresponding operations on the outputs of the original functions. Understanding these operations is essential when studying how modifications to functions affect their behavior and properties.

Domain Determination

Determining the domain of a function involves identifying all input values for which the function is defined. When performing operations on functions, their domains may be restricted by issues such as division by zero or taking even roots of negative numbers. Analyzing the restrictions on each function individually and then collectively ensures that the resulting function is properly defined wherever it is intended to operate.

Function Composition

Function composition is the process of applying one function to the results of another, which is represented by f(g(x)) or g(f(x)). This operation requires careful attention to the domains of both functions because the output of the inner function must fall within the domain of the outer function. Composition is a fundamental concept in algebra and calculus, as it describes how one process can be nested within another.

Function Evaluation and Substitution

Function evaluation involves replacing the variable in a function with a specific number or expression. This procedure, often called substitution, is used to compute the value of the function for various inputs or to simplify expressions involving functions. It is a basic yet powerful technique that underpins many more advanced topics in algebra and calculus.

Radical Functions

Radical functions, such as those involving square roots, produce outputs based on an even root of an expression. Special care must be taken with these functions because the expression under the radical (the radicand) must be non-negative to produce real number outputs. Understanding the restrictions and properties of radical functions is crucial for accurately evaluating and manipulating these functions in different mathematical contexts.

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