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Chapter 1: Problem 36

Let \(f(x)=2 x^{2}+x\). Find the following. (a) \(f(3)\) (b) \(f(2 x)\) (c) \(f(1+x)\) (d) \(f\left(\frac{1}{x}\right)\) (e) \(\frac{1}{f(x)}\)

Short Answer

Expert verified
(a) \(f(3) = 21\)\n (b) \(f(2x) = 8x^2 + 2x\)\n (c) \(f(1+x) = 2x^2 + 5x + 3\)\n (d) \(f(1/x) = 2/x^2 + 1/x\)\n (e) \(1/f(x) = 1/(x*(2x + 1))\)

Step by step solution

01

Substitute x = 3 into the function \(f(x)\)

Substitute 3 for x in the equation \(f(x) = 2x^2 + x\). So, \(f(3) = 2*(3)^2 + 3\). This simplifies to \(f(3) = 18 + 3 = 21\).
02

Substitute x = 2x into the function \(f(x)\)

Substitute 2x for x in the function \(f(x) = 2x^2 + x\). So, \(f(2x) = 2*(2x)^2 + 2x\). This simplifies to \(f(2x) = 8x^2 + 2x\).
03

Substitute x = (1+x) into the function \(f(x)\)

Substitute (1+x) for x in the function \(f(x) = 2x^2 + x\). So, \(f(1+x) = 2*(1+x)^2 + (1+x)\). Expand and simplify to get \(f(1+x) = 2x^2 + 4x + 2 + x + 1 = 2x^2 + 5x + 3\).
04

Substitute x = 1/x into the function \(f(x)\)

Substitute (1/x) for x in the function \(f(x) = 2x^2 + x\). So, \(f(1/x) = 2*(1/x)^2 + 1/x\). Simplify to get \(f(1/x) = 2/x^2 + 1/x\).
05

Find the inverse of the function \(f(x)\)

To find \(1/f(x)\), let's write \(f(x) = 2x^2 + x\) as \(f(x) = x(2x + 1)\). So, \(1/f(x) = 1/(x*(2x + 1))\).

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Key Concepts

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

Polynomial Functions
Polynomial functions are expressions consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents of variables. They are one of the most fundamental concepts in algebra and can describe a vast range of mathematical situations. For example, a polynomial function like the one given in the exercise is expressed as:
  • The function \(f(x) = 2x^2 + x\) is a quadratic polynomial because the highest power of \(x\) is 2.
  • Polynomial functions can be used to model various situations such as projectile paths, area calculations, or any situation with a parabolic relationship.
  • The coefficients in the polynomial function (like 2 and 1 in the function \(f(x)\)) determine the shape and position of its graph.
Substitution Method
The substitution method involves replacing the variable in a function with expressions or specific values to evaluate the function. This technique simplifies understanding how changes in the input affect the output.
In our example, the method is used to find values like \(f(3)\) or \(f(2x)\), where:
  • For \(f(3)\), we substitute \(3\) into every \(x\) in the function, leading to \(f(3) = 2 \times (3)^2 + 3\).
  • For \(f(2x)\), the entire expression \(2x\) is substituted for every \(x\) in the function, resulting in recalculating to form \(f(2x) = 8x^2 + 2x\).
This method is essential for evaluating functions at specific points or using algebraic transformations for broader applications.
Inverse Function
An inverse function essentially "undoes" the work of the original function. However, this concept slightly alters when talking about reciprocal functions, like \(1/f(x)\). Here, we are finding a function that gives us the reciprocal of the original function's output rather than its inverse.
The exercise asks for \(1/f(x)\), calculated as:
  • First, express \(f(x)\) in a simpler product form: \(f(x) = x(2x+1)\).
  • Then, the reciprocal \(1/f(x)\) follows as \(1/(x(2x+1))\).
It's helpful because reciprocal functions can reveal behaviors in systems where inverse relations are considered, like in fractions or rates.
Quadratic Functions
Quadratic functions are a type of polynomial function where the highest exponent of the variable is 2, forming a parabolic graph when visualized. Our specific example, \(f(x) = 2x^2 + x\), is a quadratic function.
Important features of quadratic functions include:
  • The standard form: \( ax^2 + bx + c\). Here, \(a = 2\), \(b = 1\), and \(c = 0\).
  • The parabola's vertex, representing the function's minimum or maximum value depending on the coefficient \(a\).
  • The direction of the parabola (upwards or downwards) is determined by the sign of \(a\). Since \(a = 2\) is positive, the parabola opens upwards.
  • The axis of symmetry is located at \(x = -\frac{b}{2a}\), helping to understand the symmetry of the function's graph.
  • Solutions to the quadratic function, or "roots," can be found using the quadratic formula when set equal to zero.
Quadratic functions model numerous physical phenomena, including motion and optimization problems.

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Most popular questions from this chapter

For Problems 7 through 9 determine whether the relationship described is a function. If the relationship is a function, (a) what is the domain? the range? (b) is the function 1 -to- 1 ? $$ \begin{array}{ll} \text { Input } & \text { Output } \\ \hline \sqrt{2} & 2 \\ \sqrt{3} & 3 \\ \sqrt{5} & 5 \\ \sqrt{6} & 6 \end{array} $$

(a) A bead maker has a collection of wooden spheres 2 centimeters in diameter and is making beads by drilling holes through the center of each sphere. The length of the bead is a function of the diameter of the hole he drills. Find a formula for this function. If you are stuck, begin by trying to express half the length of the bead as a function of the radius of the hole drilled. (b) More generally, suppose he works with spherical beads of radius \(\mathrm{R}\). Again, the length of the bead is a function of the diameter of the hole he drills. Find a formula for this function. In the following problems, demonstrate your use of the portable strategies for problem solving described in this chapter. What simpler questions are you asking yourself? What concrete example can you give to convince your friends and relatives that you are right? Write this up clearly, so a reader can follow your train of thought easily.

The volume of a sphere and the surface area of a sphere are both functions of the sphere's radius. The volume function is given by \(V(r)=\frac{4}{3} \pi r^{3}\) and the surface area function is given by \(S(r)=4 \pi r^{2}\). (a) If the radius of a sphere is doubled, by what factor is the volume multiplied? The surface area? (b) Which results in a larger increase in surface area: increasing the radius of a sphere by 1 unit or increasing the surface area by 12 units? Does the answer depend upon the original radius of the sphere? Explain your reasoning completely. (It may be useful to check your answer in a specific case as a spot check for errors.) (c) In order to double the surface area of the sphere, by what factor must the radius be multiplied? (d) In order to double the volume of the sphere, by what factor must the radius be multiplied?

There are infinitely many prime numbers. This has been known for a long time; Euclid proved it sometime between 300 B.C. and 200 B.C. \({ }^{6}\) Number theorists (mathematicians who study the theory and properties of numbers) are interested in the distribution of prime numbers. Let \(P(n)=\) number of primes less than or equal to \(n\), where \(n\) is a positive number. Is \(P(n)\) a function? Explain.

A gardener has a fixed length of fence that she will use to fence off a rectangular chili pepper garden. Express the area of the garden as a function of the length of one side of the garden. If you have trouble, reread the "Portable Strategies for Problem Solving" listed in this chapter. We've also included the following advice geared specifically toward this particular problem. Give the length of fencing a name, such as \(L\). (We don't know what \(L\) is, but we know that it is fixed, so \(L\) is a constant, not a variable.) \- Draw a picture of the garden. Call the length of one side of the fence \(s\). How can you express the length of the adjacent side in terms of \(L\) and \(s ?\) \- What expression gives the area enclosed by the fence?
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