Question

Find \(g(3)\) where \(g(x)=\left\\{\begin{array}{ll}3 x^{2}-7 x & \text { if } \quad x<0 \\ 5 x-9 & \text { if } \quad x \geq 0\end{array}\right.\) Find \(g(3)\) where $$ g(x)=\left\\{\begin{array}{ll} 3 x^{2}-7 x & \text { if } \quad x<0 \\ 5 x-9 & \text { if } \quad x \geq 0 \end{array}\right. $$

Short answer

6

Step-by-step solution

Identify the correct piece of the piecewise function

First, determine which piece of the piecewise function to use. Since we need to find \(g(3)\) and \(3 \geq 0\), we use the piece \(5x - 9\).

Substitute the value into the function

Substitute \(x = 3\) into the correct piece of the function: \(g(x) = 5x - 9\).

Calculate \(g(3)\)

Now, calculate \(g(3)\) by substituting \(x = 3\): \(g(3) = 5(3) - 9\).

Perform the arithmetic

Multiply 5 by 3 to get 15, and then subtract 9: \(g(3) = 15 - 9 = 6\).

Key concepts

Understanding Function Evaluation

Function evaluation is an essential concept in algebra. It involves finding the output of a function given a specific input value. Functions are mathematical relations that assign a unique output for each input. To evaluate a function at a specific point, you substitute the input value into the function's formula. For instance, to evaluate the function \(g(x) = 5x - 9\) at \(x = 3\), you simply replace \(x\) with 3 and perform the arithmetic. This process helps you determine what the function equals at that point. Function evaluation is fundamental for understanding more complex mathematical concepts and is widely used in calculus, physics, and engineering.

Basics of Algebra in Piecewise Functions

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of piecewise functions, algebra helps us define different expressions based on specific conditions. A piecewise function can have multiple expressions, each defined over a different interval. For example, a piecewise function like \(g(x)\) can be represented as \[g(x)= \begin{cases} 3x^2 - 7x & \text {if} \ x<0 \5x - 9 & \text {if} \ xeq 0 \end{cases}\]. Here, algebra allows us to create these expressions and evaluate them based on the input value. The key algebraic steps in solving piecewise functions include:

• Identifying the correct piece based on the input value
• Substituting the input value
• Performing arithmetic operations to get the output

By understanding and applying these algebraic rules, you can efficiently solve piecewise function problems.

Understanding Piecewise-Defined Functions

Piecewise-defined functions are functions defined by different expressions based on the input value. They are useful for representing situations where a rule changes depending on the input. For example, the function \(g(x) = \begin{cases} 3x^2 - 7x & \text {if} \ x<0 \ 5x - 9 & \text {if} \ x \geq 0 \end{cases}\) means that for inputs less than zero, we use the expression \(3x^2 - 7x\), and for inputs greater than or equal to zero, we use the expression \(5x - 9\). Understanding piecewise functions relies on:

• Knowing which piece to use based on the given condition
• Applying the relevant piece of the function
• Evaluating the function accordingly

Let's evaluate \(g(3)\) step-by-step. Since \(3 \geq 0\), we use the second piece of the function: \(g(x) = 5x - 9\). Substitute \(x = 3\) into this expression: \(g(3) = 5(3) - 9 = 15 - 9 = 6\). Thus, \(g(3) = 6\). Piecewise functions are important in various fields, including programming, physics, and economics, where different rules apply under varying conditions.