Common Functions: Videos & Practice Problems

Understanding common functions is essential for mastering mathematical concepts. This summary explores several key functions, their definitions, and their characteristics, including domain and range.

The constant function is defined as f(x) = c, where c is any constant number. For example, if f(x) = 2, the graph is a horizontal line at y = 2. The domain of a constant function is all real numbers, represented as (−∞, +∞), since any value of x can be input. However, the range is limited to the constant value, so in this case, it is simply {2}.

The identity function is expressed as f(x) = x. This function outputs the same value as the input, meaning if you input -1, the output is -1, and if you input 50, the output is 50. Both the domain and range for the identity function are all real numbers, (−∞, +∞).

The square function, defined as f(x) = x², produces a parabolic graph. The domain is all real numbers, (−∞, +∞), since you can square any real number. However, the range is restricted to non-negative values, [0, +∞), as the output cannot be negative.

The cube function is given by f(x) = x³. This function includes all real numbers for both the domain and range, (−∞, +∞), as cubing any real number yields a real number, whether positive or negative.

The square root function is represented as f(x) = √x. This function has restrictions: the domain is limited to non-negative values, [0, +∞), because you cannot take the square root of a negative number. The range is also [0, +∞), as the output is always non-negative.

Lastly, the cube root function is defined as f(x) = ∛x. Similar to the cube function, the domain and range for the cube root function encompass all real numbers, (−∞, +∞), allowing for both negative and positive inputs and outputs.

Familiarity with these functions and their properties is crucial for further studies in mathematics, as they frequently appear in various mathematical contexts.

Understanding the slope-intercept form of a linear equation is essential for graphing and analyzing lines. The slope-intercept form is expressed as y = mx + b, where m represents the slope of the line and b denotes the y-intercept.

The slope, m, is calculated using the formula for rise over run, which can be expressed as m = \frac{\Delta y}{\Delta x}. This means you determine how much the line rises (the change in y) for a given run (the change in x) between two points on the line. For example, if the rise is 2 and the run is 1, the slope would be m = \frac{2}{1} = 2.

The y-intercept, b, is the point where the line crosses the y-axis, which occurs when x = 0. For instance, if the graph crosses the y-axis at the point (0, 3), then b = 3. If it crosses at (0, -3), then b = -3.

To write the equation of a line in slope-intercept form, simply substitute the values of m and b into the equation. For example, if the slope is 2 and the y-intercept is 3, the equation becomes y = 2x + 3.

In another example, if the slope is 1 (calculated as a rise of 1 and a run of 1) and the y-intercept is -3, the equation would be y = 1x - 3, which can be simplified to y = x - 3.

By mastering the slope-intercept form, you can easily describe and graph linear equations, making it a fundamental skill in algebra.

Quadratic functions are a specific type of polynomial function characterized by their degree of 2. The standard form of a quadratic function is expressed as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are real numbers, and \( a \neq 0 \). The graph of any quadratic function takes the shape of a parabola, which can open either upward or downward depending on the sign of \( a \).

The vertex of a parabola is a crucial feature, representing either the highest or lowest point on the graph. For a parabola that opens upward, the vertex is the minimum point, while for one that opens downward, it is the maximum point. The vertex is denoted as an ordered pair, such as \( (h, k) \), where \( h \) and \( k \) are the coordinates of the vertex.

Another important aspect of quadratic functions is the x-intercepts, which are the points where the graph crosses the x-axis. A quadratic function can have either one or two x-intercepts, but never more. The y-intercept, on the other hand, is where the graph intersects the y-axis, and it can be found by evaluating \( f(0) \).

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, always passing through the vertex. It can be expressed as \( x = h \), where \( h \) is the x-coordinate of the vertex.

When analyzing the domain and range of quadratic functions, the domain is always all real numbers, represented as \( (-\infty, \infty) \). The range, however, depends on whether the vertex is a minimum or maximum. If the vertex is a minimum, the range extends from the y-coordinate of the vertex to infinity. Conversely, if the vertex is a maximum, the range extends from negative infinity to the y-coordinate of the vertex.

Additionally, understanding the intervals of increase and decrease is essential. A parabola increases on the interval from negative infinity to the x-coordinate of the vertex and decreases from the x-coordinate of the vertex to positive infinity. This behavior is a direct result of the parabolic shape.

Quadratic functions can also be expressed in vertex form, which is particularly useful for graphing. The vertex form is given by \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. This form allows for easier identification of the vertex and the direction in which the parabola opens.

Polynomial functions are a broad category of mathematical expressions characterized by their use of positive whole number exponents. A polynomial function can be expressed in the form \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_n \) represents the leading coefficient, \( n \) is the degree of the polynomial, and the terms are arranged in descending order of power. For example, in the polynomial \( f(x) = 5x^3 - x^2 - 6x + 4 \), the degree is 3, and the leading coefficient is 5.

To determine if a given expression is a polynomial function, it is essential to check that all exponents are positive whole numbers. For instance, \( f(x) = 2x^{1/2} + 3 \) is not a polynomial function due to the fractional exponent. However, \( f(x) = -\frac{2}{3}x^4 + 4 \) is valid since the exponent is a positive whole number, even though it has a fractional coefficient.

The graphs of polynomial functions exhibit two key characteristics: they are smooth and continuous. This means that the graph does not have any corners or breaks. For example, a quadratic function, which is a specific type of polynomial function, maintains this smooth and continuous nature. In contrast, graphs that display sharp corners or breaks do not represent polynomial functions.

Additionally, the domain of polynomial functions is always all real numbers, extending from negative infinity to positive infinity. This property is consistent across all polynomial functions, including quadratics. Understanding these foundational aspects of polynomial functions is crucial for further exploration in algebra and calculus.

Rational functions are formed by placing one polynomial over another in a fraction, represented as \( \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials of any degree. Understanding rational functions involves recognizing that the denominator cannot equal zero, which leads to domain restrictions. For example, if the denominator is \( x - 1 \), then \( x \) cannot equal 1, as this would make the denominator zero. This restriction is crucial for defining the domain of the function, which can be expressed in set notation as \( \{ x \, | \, x \neq 1 \} \).

To find the domain of a rational function, set the denominator equal to zero and solve for \( x \). For instance, for the function \( \frac{3}{3x - 12} \), setting the denominator to zero gives \( 3x - 12 = 0 \). Solving this results in \( x = 4 \), indicating that the domain is all real numbers except \( x = 4 \). Similarly, for \( f(x) = \frac{x + 5}{x^2 - 25} \), setting the denominator \( x^2 - 25 = 0 \) leads to \( x = 5 \) and \( x = -5 \), thus the domain restriction is \( x \neq 5 \) and \( x \neq -5 \).

Another important aspect of rational functions is simplifying them to their lowest terms. This involves factoring both the numerator and the denominator and canceling any common factors. For example, in the function \( \frac{3}{3x + 12} \), the denominator can be factored to \( 3(x + 4) \), allowing the common factor of 3 to be canceled, resulting in \( \frac{1}{x + 4} \). In the case of \( f(x) = \frac{x + 5}{x^2 - 25} \), the denominator factors to \( (x + 5)(x - 5) \), allowing the \( x + 5 \) terms to cancel, simplifying the function to \( \frac{1}{x - 5} \).

It is essential to determine the domain before simplifying the function. Simplifying first may lead to missing restrictions, as seen in the previous example where the domain restriction of \( x \neq -5 \) would be overlooked if only the simplified function \( \frac{1}{x - 5} \) were considered. Thus, always find the domain of the original rational function before proceeding to simplify.

Exponential functions are a fundamental concept in mathematics, distinct from polynomial functions. A common example of an exponential function is f(x) = 2^x, where the base (2) is a constant and the exponent (x) is a variable. Understanding the characteristics of exponential functions is crucial for evaluating and graphing them effectively.

For an expression to qualify as an exponential function, the base must meet specific criteria: it must be a constant, positive, and not equal to 1. For instance, in the function f(x) = (2/3)^x, the base is 2/3, which is constant, positive, and not equal to 1, confirming it as an exponential function. Conversely, f(y) = 1^y does not qualify because the base is 1, which violates the criteria. Another example, f(x) = 10^(x+1), is valid since the base (10) is constant, positive, and not equal to 1, while the exponent x + 1 contains a variable.

Evaluating exponential functions involves substituting specific values for the variable. For example, to evaluate f(x) = 2^x at x = 4, you compute f(4) = 2^4 = 16. When dealing with negative exponents, such as f(-3) = 2^{-3}, it translates to 1/(2^3) = 1/8. For non-integer values, like x = 3.14, a calculator is often necessary. You would input 2, use the caret key for exponentiation, and then enter 3.14 to find f(3.14) ≈ 8.815. Similarly, for larger exponents, such as x = 12, you would calculate f(12) = 2^{12} = 4096 using a calculator.

In summary, recognizing the structure of exponential functions and mastering their evaluation is essential for further studies in mathematics, particularly in fields involving growth and decay, such as biology, finance, and physics.

When solving polynomial equations, such as \( x^3 = 216 \), the goal is to isolate \( x \) by performing the reverse operation. In this case, taking the cube root of both sides yields \( x = \sqrt[3]{216} \). However, when the variable \( x \) is in the exponent, as in \( 2^x = 8 \), we need to determine how many times 2 must be multiplied by itself to equal 8, which gives us \( x = 3 \). For more complex equations like \( 2^x = 216 \), instead of multiplying repeatedly, we can use logarithms to simplify the process.

The logarithm is the inverse operation of exponentiation. To isolate \( x \) in the equation \( 2^x = 216 \), we take the logarithm of both sides, specifically using base 2 to match the base of the exponent. This results in \( \log_2(2^x) = \log_2(216) \). Since the logarithm and the exponential have the same base, they cancel out, leading to \( x = \log_2(216) \). This logarithmic form expresses the power to which the base (2) must be raised to yield 216.

Understanding logarithmic and exponential forms is crucial. For instance, converting from exponential to logarithmic form involves starting with the base of the exponent. For the equation \( 3^x = 81 \), we convert it to logarithmic form as \( \log_3(81) = x \). Conversely, to convert from logarithmic to exponential form, such as \( x = \log_4(64) \), we start with the base (4) and express it as \( 4^x = 64 \).

To practice these conversions, consider \( x = \log_5(800) \). The equivalent exponential form is \( 5^x = 800 \). Similarly, for \( \log_2(16) = 4 \), the exponential form is \( 2^4 = 16 \), confirming the accuracy of our conversion. Lastly, for \( 10^x = 45100 \), the logarithmic form is \( x = \log_{10}(45100) \), which is often referred to as the common logarithm and can simply be written as \( \log(45100) \). This common logarithm has a dedicated button on calculators for easy evaluation.

By mastering the relationship between logarithmic and exponential forms, students can simplify complex equations and enhance their problem-solving skills in mathematics.

Graphing logarithmic functions can initially seem daunting, but it closely mirrors the process of graphing exponential functions due to their inverse relationship. For instance, when working with the logarithmic function log2(x), we can derive ordered pairs by substituting values for x. Starting with x = 1, we find that log2(1) = 0, giving us the point (1, 0). Next, for x = 2, log2(2) = 1, resulting in the point (2, 1).

These points can be compared to the corresponding exponential function f(x) = 2x, where the points (0, 1) and (1, 2) can be flipped to yield the points for the logarithmic function. This flipping of coordinates is a fundamental property of inverse functions, allowing us to easily generate the necessary points for our logarithmic graph.

As we plot these points, we observe that the graph approaches the y-axis but never touches it, indicating a vertical asymptote at x = 0. This behavior is consistent across all logarithmic functions, where the domain is all positive real numbers ((0, ∞)) and the range is all real numbers ((−∞, ∞)).

When comparing the shapes of the graphs, one can visualize the exponential function resembling a lowercase "e" and the logarithmic function resembling an extended lowercase "r". This visual cue can help in recalling the general shapes of these functions.

Furthermore, the direction of the logarithmic graph is influenced by the base b. If b > 1, the graph will be increasing, while if 0 < b < 1, the graph will be decreasing. This characteristic is similar to that of exponential functions, reinforcing the connection between these two types of functions.

In summary, understanding the relationship between logarithmic and exponential functions allows for a more intuitive approach to graphing. By leveraging the properties of inverse functions, one can easily derive the necessary points and understand the overall behavior of logarithmic graphs.

In this exploration of functions, we focus on two specific mathematical expressions: \( f(x) = 3^x \) and \( g(x) = \log_3(x) \). These functions are inverses of each other, as the logarithm serves as the inverse of an exponential function. To understand their behavior, we will graph both functions and analyze their key characteristics.

Starting with the exponential function \( f(x) = 3^x \), we can calculate several points to plot. For \( x = 0 \), we find \( f(0) = 3^0 = 1 \). For \( x = 1 \), \( f(1) = 3^1 = 3 \), and for \( x = 2 \), \( f(2) = 3^2 = 9 \). As \( x \) takes on negative values, the function yields fractions: \( f(-1) = 3^{-1} = \frac{1}{3} \) and \( f(-2) = 3^{-2} = \frac{1}{9} \). This indicates that as \( x \) decreases, \( f(x) \) approaches zero but never actually reaches it, demonstrating a horizontal asymptote at \( y = 0 \).

Next, we turn to the logarithmic function \( g(x) = \log_3(x) \). To graph this function, we can utilize the points derived from the exponential function by swapping the \( x \) and \( y \) coordinates. Thus, the points become: \( (1, 0) \), \( (3, 1) \), and \( (9, 2) \) for positive values, while for negative values, we have \( \left(\frac{1}{3}, -1\right) \) and \( \left(\frac{1}{9}, -2\right) \). This function also approaches a vertical asymptote at \( x = 0 \), indicating that the logarithm is undefined for non-positive values.

When analyzing the domains and ranges of these functions, we find that the domain of \( f(x) = 3^x \) encompasses all real numbers, \( (-\infty, \infty) \). Correspondingly, the range of this function is \( (0, \infty) \), as it never reaches zero. Conversely, the domain of \( g(x) = \log_3(x) \) is restricted to positive values, \( (0, \infty) \), while its range spans all real numbers, \( (-\infty, \infty) \). This relationship between the domains and ranges of inverse functions is a fundamental concept in mathematics.

In summary, the graphs of \( f(x) = 3^x \) and \( g(x) = \log_3(x) \) illustrate their inverse relationship, characterized by their respective asymptotes and the swapping of their domains and ranges. Understanding these properties enhances our grasp of exponential and logarithmic functions, which are essential in various mathematical applications.