Problem 77 - Elementary and Intermediate Algebra

Question

Let \(h(t)=\sqrt{t+3}\) and \(k(t)=t-5 .\) Find each of the following. $$ (h \circ k)(18) $$

Step-by-Step Solution

Answer

(h \circ k)(18) = 4.

Step 1: Understand the Composition

The composition of functions \((h \circ k)(t)\) means we first apply the function \(k(t)\) and then apply the function \(h(t)\) to the result of \(k(t)\). Therefore, \((h \circ k)(t) = h(k(t))\).

Step 2: Evaluate \(k(18)\)

Substitute \(t = 18\) into the function \(k(t) = t - 5\). This gives: \[ k(18) = 18 - 5 = 13 \]

Step 3: Evaluate \(h(k(18))\)

Now substitute the result from the previous step into the function \(h(t) = \sqrt{t+3}\). This means substituting \(13\) into \(h(t)\).\[ h(13) = \sqrt{13 + 3} = \sqrt{16} \]

Step 4: Calculate \(\sqrt{16}\)

Since \(\sqrt{16} = 4\), this is the value for \(h(13)\).

Key Concepts

Understanding the Square Root Function

The square root function is a special type of mathematical operation that reverses the process of squaring a number. When you see the square root sign, it means you are looking for a number that, when multiplied by itself, gives the original number beneath the square root symbol.For example,

• the square root of 16 is 4 because when you multiply 4 by 4, you get 16.
• Similarly, the square root of 9 is 3, as 3 times 3 equals 9.
• Did you know that the square root of 1 is just 1? That's because 1 multiplied by itself is still 1.

The function we are using here, \(h(t) = \sqrt{t+3}\), means that first you add 3 to your input \(t\), and then you find the square root of that sum.It’s key to make sure the expression inside the square root results in a non-negative number, as the square root of a negative number is not defined in the set of real numbers.Understanding this function is crucial when combined with other functions in processes such as function composition.

Mastering Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. Imagine it as feeding a number into a machine that follows a set of instructions to produce a result.Let's take the function \(k(t) = t - 5\). To evaluate this function for \(t = 18\):

• You simply substitute 18 for \(t\) in the function.
• So you calculate \(18 - 5\), which equals 13.

Function evaluation is a fundamental skill in math because it helps us determine how different inputs change the output.As you saw in the exercise, evaluating functions step-by-step helps check your understanding and ensures accuracy in more complex problems.

Math Problem Solving with Function Composition

Function composition is like chaining together two or more functions, where the output of one function becomes the input of another. This comes in handy when you need to perform complex calculations in a sequence.In the exercise,

• the function \((h \circ k)(t)\) symbolizes: first apply \(k(t)\), then apply \(h(t)\) to that result.
• This is done step by step to maintain accuracy and logic in your calculations.
• First, you evaluate \(k(t)\), then you use the result to evaluate \(h(t)\). This stepwise approach simplifies how you manage multiple functions and the transformations between them.

Of course, careful execution is important. By fully grasping each individual function before composing them, you make sure your problem-solving process is as smooth as possible.