Lessons 33–34 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
34
The exponential function and its graph
Textbook, Part 2: pp. 53–54
GoalKnow the properties of powers with real exponents, state the properties and graph of the exponential function and compare powers.
For a > 0, a ≠ 1, f(x) = aˣ is called an exponential function. For real exponents aˣ · aʸ = aˣ⁺ʸ, aˣ : aʸ = aˣ⁻ʸ, (aˣ)ʸ = aˣʸ and (ab)ˣ = aˣbˣ hold. The domain is (−∞, +∞), the range is (0, +∞); a⁰ = 1, so the graph passes through (0, 1) and never meets the Ox axis. If a > 1 the function is increasing: x₁ < x₂ ⇒ aˣ¹ < aˣ². If 0 < a < 1 it is decreasing: x₁ < x₂ ⇒ aˣ¹ > aˣ². By this property powers with equal bases are compared: for a > 1 the larger exponent gives the larger power, for 0 < a < 1 the opposite. For equal exponents and different bases, if 0 < a < b then aˣ < bˣ for x > 0 and aˣ > bˣ for x < 0.
Worked examples
Compare 2^√2 and 2^(7/5). The base 2 > 1, so the function is increasing; √2 ≈ 1.414 > 7/5 = 1.4, hence 2^√2 > 2^(7/5).
(1/2)^(−3) = 2³ = 8; y = (1/2)ˣ is decreasing, its graph passes through (0, 1), with y = 8 at x = −3 and y = 1/8 at x = 3.
Class activity
Make a table of y = 2ˣ and y = (1/2)ˣ for x = −3, −2, −1, 0, 1, 2, 3. Draw both graphs in one coordinate plane and observe that they are symmetric in the Oy axis.
Practice
1
State the domain and range of y = aˣ.
Domain (−∞, +∞), range (0, +∞).
2
Which is larger: 3^(−1/2) or 3^(−2/3)?
3^(−1/2)
3
Find the value of y = (1/2)ˣ at x = −3.
8
4
Why do we not allow a = 1 as the base of an exponential function?
Because 1ˣ = 1 for all x: a constant function, neither increasing nor decreasing.