Investigating exponential and logarithmic functions · Grade 12 · 4 units
In Investigating exponential and logarithmic functions (Grade 12 · 4 units) you practise 7 skills: The domain · Where the graph meets the axes · Increasing, decreasing and turning points · Where a function is positive or negative · Even and odd functions · Asymptotes · Matching graphs. Here we investigate the function x times e to the power of x. The domain is every number, because a power of e is always defined. When x is 0 the function is 0 too, so the graph passes through the origin. That is the only point where it meets the axes, because a power of e is never 0.
By the product rule, the derivative is x plus 1, times e to the power of x. The power is always positive, so the derivative is 0 only when x is minus 1. The sign chart of the derivative shows that it is negative to the left of minus 1 and positive to the right. So the function decreases up to minus 1 and increases after it, and it has a minimum where x is minus 1. The minimum value is minus 1 over e, about minus 0.37. When x gets very small, the product gets close to 0 and never reaches it, so the x axis is a horizontal asymptote on the left. When x gets very large, the function rises without end. With all of this, sketch the graph: it hugs the axis from below on the left, dips to its minimum, and then rises steeply through the origin.
Tip: A power of e is always positive, so it never changes the sign of a product.
What you learn
The domain
For example: What is the domain of f(x) = e^(x² + 5) ?
- x > 0
- x ≠ 0
- x ≥ 0
- Every number
Answer and solution
The answer: Every number
A power of e is defined for every number, whatever is in the power. So the domain is every number.
Where the graph meets the axes
For example: Here is the function: f(x) = (x − 6) × eˣ At which point does its graph cross the vertical axis?
Answer and solution
The answer: (0, −6)
Put in 0: f(0) = −6 × 1 = −6 , so the point is (0, −6) .
Increasing, decreasing and turning points
For example: Here is the function: f(x) = (x + 4) × eˣ What is the value of x at its minimum?
Answer and solution
The answer: −5
By the product rule, f′(x) = (x + 5) × eˣ . The power is always positive, so the derivative is 0 only at x = −5 . To the left of it the derivative is negative, and to the right it is positive. So there is a minimum there.
Where a function is positive or negative
For example: Here is an inequality: (x + 5) × eˣ > 0 For which values of x is it true?
Answer and solution
The answer: x > −5
The power eˣ is always positive, so the sign is the sign of the bracket: x + 5 > 0 , that is x > −5
Even and odd functions
For example: Here is the function: f(x) = e^(−2x²) Is it even, odd or neither?
- Odd
- Neither even nor odd
- Even
Answer and solution
The answer: Even
Put in minus x: f(−x) = e^(−2x²) . This is the function itself, so it is even.
Asymptotes
For example: Here is the function: f(x) = ln(x² − 25) For which values of x does its graph have a vertical asymptote?
Answer and solution
The answer: x = −5 or x = 5
What is inside the log is 0 when x² = 25 , that is at the ends of the domain. So there are two asymptotes: x = −5 and x = 5 .
Matching graphs
For example: Here is the function: y = ln x Which of these graphs is its graph? Graph A:
Graph B: Graph C: Graph D:- Graph A
- Graph C
- Graph D
- Graph B
Answer and solution
The answer: Graph B
The function is defined only for positive numbers, is increasing, and crosses the horizontal axis at (1, 0). That is graph B.
Not on the site yet
Parts of this topic that are not built yet: The second derivative and points of inflection.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).