Learn All of Functions in only 40 Minutes! (ultimate study guide)

The 10 Essentials of Functions

The video reviews ten foundational ideas about functions, using equations, tables, graphs, and worked examples.

1. What a Function Is

• A function is a relationship in which every allowed input corresponds to exactly one output.
• For example, (y = 2x^2 - 5) is a function: each (x) gives one (y).
• The relation (x = y^2) is not a function of (x), because (x = 4) corresponds to both (y = 2) and (y = -2).
• On a graph, use the vertical line test: if any vertical line crosses the graph more than once, the relation is not a function.
• Function notation names a function and shows its input. For example, (f(x) = 2x^2 - 5), and (f(-2) = 3) means the output is 3 when the input is (-2). The notation (f(x)) does not mean (f) multiplied by (x).

2. Domain and Range

• The domain is the set of allowed input values, usually described as possible (x)-values.
• The range is the set of possible outputs, usually described as possible (y)-values.
• To find them:
  • From an equation, identify which inputs produce real outputs and which outputs are possible.
  • From a graph, consider how far the graph extends horizontally for the domain and vertically for the range.
• For (f(x) = x^2 + 1), any real (x) is allowed, so the domain is all real numbers. Since (x^2 \geq 0), the output is always at least 1, so the range is (y \geq 1).
• The square-root function (f(x) = \sqrt{x}) is another example of a restricted domain: its real-valued inputs must satisfy (x \geq 0).

3. Common Function Graphs

The video introduces several standard graph shapes to recognize:

• Power functions: (x) is linear, (x^2) is a parabola, and (x^3) has an S-shape.
• Square root: begins at the origin and extends to the right.
• Reciprocal: (1/x) has two branches and approaches the axes as asymptotes.
• Sine and cosine: periodic waves that oscillate between (-1) and (1); cosine resembles a horizontally shifted sine graph.
• Exponential: a function such as (2^x) grows rapidly to the right and approaches (y = 0) to the left.
• Logarithmic: a function such as (\log_2 x), the inverse of (2^x), has a vertical asymptote and grows more slowly as (x) increases.

4. Transformations of Functions

A function’s graph can be shifted, stretched, compressed, or reflected. The video describes a general transformed form as

[ g(x) = a\,f\bigl(k(x-d)\bigr) + c. ]

For this form:

• Vertical shift (c): a positive value shifts the graph up; a negative value shifts it down.
• Horizontal shift (d): a positive value shifts the graph right; a negative value shifts it left.
• Vertical factor (a):
  • If (|a| > 1), the graph is stretched vertically.
  • If (0 < |a| < 1), the graph is compressed vertically.
  • A negative (a) reflects the graph across the (x)-axis.
• Horizontal factor (k):
  • If (|k| > 1), the graph is compressed horizontally by a factor of (1/|k|).
  • If (0 < |k| < 1), the graph is stretched horizontally by a factor of (1/|k|).
  • A negative (k) reflects the graph horizontally.

In the worked example, the video starts with (f(x) = x^3), then applies a vertical compression, a shift right by 2, and a shift down by 1. It demonstrates these changes by transforming points on the parent graph.

5. Inverse Functions

• An inverse function reverses the action of the original function: if (f(a) = b), then (f^{-1}(b) = a).
• The notation (f^{-1}) indicates an inverse, not a reciprocal or a power of (-1).
• To find an inverse from an equation:
  1. Write the function using (y).
  2. Swap (x) and (y).
  3. Solve for (y).
• The graphs of a function and its inverse are reflections across (y = x); their corresponding points have their coordinates reversed.
• For example, the inverse of (2^x) is (\log_2 x).

6. Combining Functions

• Functions can be combined by adding, subtracting, multiplying, or dividing their outputs.
• For addition, ((f + g)(x) = f(x) + g(x)).
• The video demonstrates two approaches:
  • Graphically: combine the outputs corresponding to each input (x). For addition, add the (y)-values.
  • Algebraically: add the function expressions, simplify, and graph the resulting equation.
• In the example, a quadratic expression and a linear expression are added. The resulting quadratic can then be analyzed using features such as its intercepts and vertex.

7. Composite Functions

• A composite function uses the output of one function as the input of another.
• The notation (f(g(x))) means: first evaluate (g(x)), then use that result as the input to (f).
• To find a composite expression, substitute the entire inner function into the outer function and simplify.
• To evaluate a composite at a number, work from the inside out. For example, to find (g(f(4))), first calculate (f(4)), then use that result as the input to (g).

8. Piecewise Functions

• A piecewise function uses different formulas over different parts of its domain.
• To graph one:
  1. Graph each formula as if it were its own function.
  2. Keep only the part of each graph that satisfies that formula’s domain condition.
  3. Use a closed circle when the endpoint is included and an open circle when it is excluded.
• In the example, the function follows a linear rule for (x < 0) and a quadratic rule for (x \geq 0). Therefore, the point at (x = 0) is included on the quadratic piece, not the linear one.

9. Even and Odd Functions

• An even function satisfies (f(-x) = f(x)). Its graph is symmetric across the (y)-axis.
• An odd function satisfies (f(-x) = -f(x)). Its graph is symmetric about the origin, meaning a 180-degree rotation around the origin leaves it unchanged.
• To test a function algebraically:
  1. Replace (x) with (-x) to find (f(-x)).
  2. Simplify.
  3. Compare the result with (f(x)) to test for evenness, or with (-f(x)) to test for oddness.

10. Asymptotes

• A vertical asymptote has the form (x = a). The function’s values grow without bound or decrease without bound as (x) approaches that line.
  • For rational functions, a nonzero numerator divided by zero at (x = a) indicates a vertical asymptote, provided the expression has not been simplified to remove the discontinuity.
  • From a graph, the function approaches (+\infty) or (-\infty) near the asymptote.
• A horizontal asymptote has the form (y = b). The function approaches that value as (x) goes toward positive or negative infinity.
  • For rational functions, the video demonstrates finding the limit by dividing the terms by the highest power of (x) in the denominator and examining what remains as (x) becomes very large.
  • In the example, the function approaches (y = 0), so (y = 0) is a horizontal asymptote.

Speakers and Sources

• One unnamed presenter/instructor provides the explanations and worked examples throughout. The video title and metadata identify the channel as JensenMath.
• No other speakers or external sources are featured in the subtitles.