# Investigating exponential and logarithmic functions · Grade 12 · 4 units

> Explanation, worked examples and free practice following the Israeli curriculum.

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

- Example: What is the domain of f(x) = e^(x² + 5) ?
- Options: x > 0 · x ≠ 0 · x ≥ 0 · Every number
- Answer: Every number
- Solution: 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

- Example: Here is the function: f(x) = (x − 6) × eˣ At which point does its graph cross the vertical axis?
- Answer: (0, −6)
- Solution: Put in 0: f(0) = −6 × 1 = −6 , so the point is (0, −6) .

### Increasing, decreasing and turning points

- Example: Here is the function: f(x) = (x + 4) × eˣ What is the value of x at its minimum?
- Answer: −5
- Solution: 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

- Example: Here is an inequality: (x + 5) × eˣ > 0 For which values of x is it true?
- Answer: x > −5
- Solution: 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

- Example: Here is the function: f(x) = e^(−2x²) Is it even, odd or neither?
- Options: Odd · Neither even nor odd · Even
- Answer: Even
- Solution: Put in minus x: f(−x) = e^(−2x²) . This is the function itself, so it is even.

### Asymptotes

- Example: Here is the function: f(x) = ln(x² − 25) For which values of x does its graph have a vertical asymptote?
- Answer: x = −5 or x = 5
- Solution: 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

- Example: Here is the function: y = ln x Which of these graphs is its graph? Graph A: (Graph A: a falling curve above the horizontal axis that crosses the vertical axis at height 1 and gets close to the horizontal axis on the right.) Graph B: (Graph B: a rising curve that lies wholly to the right of the vertical axis, crosses the horizontal axis at (1, 0) and falls steeply near the vertical axis.) Graph C: (Graph C: a falling curve below the horizontal axis that crosses the vertical axis at height −1 and gets close to the horizontal axis on the left.) Graph D: (Graph D: a rising curve above the horizontal axis that crosses the vertical axis at height 1 and gets close to the horizontal axis on the left.)
- Options: Graph A · Graph C · Graph D · Graph B
- Answer: Graph B
- Solution: 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

- The second derivative and points of inflection

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 12 · 4 units topics](https://kixplo.com/il/en/math/grade-12-4u/)