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Chapter 1: Problem 11

Evaluate \(f(3), f(-1),\) and \(f(0)\). $$f(t)=\sqrt{3 t+4}$$

Short Answer

Expert verified
For the given function \(f(t) = \sqrt{3t+4}\), the values are \(f(3) = \sqrt{13}\), \(f(-1) = \sqrt{1} = 1\) and \(f(0) = \sqrt{4} = 2\)

Step by step solution

01

Substitution of t=3

Substitute \(t=3\) into the formula \(f(t) = \sqrt{3t+4}\). Hence, we get \(f(3) = \sqrt{3*3+4} = \sqrt{13}\)
02

Substitution of t=-1

Now substitute \(t=-1\) into the formula. Hence, we get \(f(-1) = \sqrt{3*(-1)+4} = \sqrt{1}\).
03

Substitution of t=0

Now, substitute \(t=0\) into the formula. Hence, we get \(f(0) = \sqrt{3*0+4} = \sqrt{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
A square root function, like the one in the exercise, is a type of function that involves finding the square root of an expression. More formally, a square root function can be expressed as \( f(x) = \sqrt{x} \). In this case, our function involves an extra term: \( f(t) = \sqrt{3t + 4} \). This function translates the basic square root function horizontally and vertically, changing the shape of its graph.

Essential characteristics include:
  • The square root function only works with non-negative numbers under the square root, because square roots of negative numbers are not defined in the real number system.
  • It is important to evaluate whether the expression inside the square root remains positive or zero, otherwise the function isn't real-valued.
Understanding how to evaluate these conditions ensures you correctly determine function values. This is essential when catching mistakes or inconsistencies when evaluating functions.
Substitution Method
The substitution method is a simple technique often used to evaluate functions. It involves replacing the variable in the function with a given number, as shown in the original exercise.

To apply this method:
  • Identify the expression for which you need to find the function's value. In our step-by-step solution, we have the expression \( f(t) = \sqrt{3t + 4} \).
  • For each specific value you substitute, replace the variable \( t \) with that number. For instance, substituting with \( t=3 \), the function becomes \( f(3) = \sqrt{3 \times 3 + 4} \).
  • Solve the resulting expression under the square root and simplify it.
This basic method helps in breaking problems into smaller, more manageable parts, and it is broadly used not just in precalculus but in all levels of math.
Precalculus
Precalculus is a course that prepares students for calculus. It includes a broad range of topics that establish a foundation for understanding the principles of calculus. Evaluating functions, like the square root function in the exercise, is a key part of precalculus.

Here鈥檚 what precalculus often covers:
  • Functions and their properties. Understanding different types of functions, such as linear, polynomial, rational, and trigonometric functions, helps build the groundwork for calculus.
  • Complex numbers, which are crucial when dealing with square roots of negative numbers or analyzing advanced equations.
  • Coordinate geometry, which helps in visualizing and solving problems involving shapes, slopes, and other properties related to graphs.
Precalculus reinforces problem-solving skills and is crucial for students aiming to understand the more advanced concepts found in calculus.

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Most popular questions from this chapter

This set of exercises will draw on the ideas presented in this section and your general math background. Show that \(|x-k|=|k-x|\), where \(k\) is any real number.

Let \(f(s)=m s+b .\) Find values of \(m\) and \(b\) such that \(f(0)=2\) and \(f(2)=-4 .\) Write an expression for the linear function \(f(s) .\) (Hint: Start by using the given information to write down the coordinates of two points that satisfy \(f(s)=m s+b .)\)

The following function gives the cost for printing \(x\) copies of a 120 -page book. The minimum number of copies printed is 200 , and the maximum number printed is 1000 $$f(x)=\left\\{\begin{array}{ll}5 x, & \text { if } 200 \leq x<500 \\\4.5 x, & \text { if } 500 \leq x<750 \\\4 x, & \text { if } 750 \leq x \leq 1000\end{array}\right.$$ (a) Find the cost for printing 400 copies of the book. (b) Find the cost for printing 620 copies of the book. (c) Which is more expensive-printing 700 copies of the book or printing 750 copies of the book?

Graph the function by hand. $$F(x)=\left\\{\begin{array}{ll} 0, & x \leq 1 \\ 2, & x>1 \end{array}\right.$$

Let \(F\) be defined as follows. $$F(x)=\left\\{\begin{array}{ll}x, & \text { if } 0 \leq x \leq 4 \\\4, & \text { if } x>4\end{array}\right.$$ Graph \(F(2 x)\)
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