Problem 58 - Precalculus : Building Concepts and Connections

Question

In Exercises \(49-66,\) let \(f(x)=x^{2}+x, g(x)=\sqrt{x},\) and \(h(x)=-3 x\) Evaluate each of the following. $$(f \circ g)(9)$$

Step-by-Step Solution

Answer

The result of \((f \circ g)(9)\) is 12.

Step 1: Compute Function \(g(9)\)

First, calculate \(g(9)\) by substituting \(x = 9\) in \(g(x) = \sqrt{x}\).\nSo, \(g(9) = \sqrt{9} = 3\).

Step 2: Compute Function \((f\circ g)(9)\)

Next, substitute \(g(9)\) into the function \(f(x) = x^2 + x\) to get \(f(g(9))\).\nSo, \(f(g(9)) = f(3) = 3^2 + 3 = 12\).

Key Concepts

Square Root Function

The square root function is a type of function that, as its name suggests, involves taking the square root of a number. It is typically represented as \(g(x) = \sqrt{x}\). In this function, the input \(x\) must be greater than or equal to zero because the square root of a negative number is not defined in the set of real numbers.

When evaluating the square root function, such as \(g(9)\), you replace \(x\) with 9 to get \(\sqrt{9}\). The square root of 9 is 3 because 3 times 3, written as \(3^2\), equals 9. Therefore, \(g(9) = 3\).

• Remember that the output of the square root function is always a non-negative number.
• Importantly, this applies regardless of whether the input is a perfect square, like 9, or not.

Quadratic Function

A quadratic function is a polynomial function of degree 2. It generally takes the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The function given in the exercise is \(f(x) = x^2 + x\), so \(a = 1\), \(b = 1\), and \(c = 0\).

Quadratic functions have a parabolic graph. This means they curve upwards like a "U" if \(a > 0\), or downwards if \(a < 0\).

When evaluating this function at a particular point, you plug in the value for \(x\). For instance, in this step-by-step example, after finding \(g(9) = 3\), we substitute 3 into \(f(x)\):

• Calculate \(f(3) = 3^2 + 3\).
• This simplifies to \(9 + 3 = 12\).

Function Evaluation

Function evaluation involves substituting a specific input value into a function to calculate its output. It is an essential skill to master in mathematics.

The process of function evaluation can be broken down into clear steps. First, identify the value of \(x\) you want to substitute into the function. Next, replace every instance of \(x\) in the function with the chosen value and perform the necessary arithmetic operations.

In the given exercise, you are evaluating a composite function \((f \circ g)(x)\). This means you first evaluate \(g(x)\), and then use that result as an input for \(f(x)\). Here's how it’s done in the exercise:

• Calculate \(g(9)\), which yields the result 3, as previously explained.
• Then, evaluate \(f(g(9))\) by finding \(f(3) = 9 + 3 = 12\).

By clearly breaking down each step, function evaluation can be straightforward and manageable, rendering it a necessary tool for tackling more complex mathematical problems.