Algebra 12 Basic Functions

LEARN Tutoring Services Twelve Basic Functions Below are the graphs of twelve functions along with domain, range, continuity, increasing/decreasing intervals, symmetry, boundedness, extrema, asymptotes and end behvior. Also please note that  is the set of integers. f ( x) = x Identity Function Domain: ( −∞, ∞ ) Range: ( −∞, ∞ ) Continuous Increasing: ( −∞, ∞ ) Decreasing: None Symmetry: origin (odd function) Not bounded Extrema: None H.A.: None V.A.: None End behavior: lim f ( x) = −∞ lim f ( x ) = ∞ f ( x) = x3 Cubing Function Domain: ( −∞, ∞ ) Range: ( −∞, ∞ ) Continuous Increasing: ( −∞, ∞ ) Decreasing: None Symmetry: origin (odd function) Not bounded Extrema: None H.A.: None V.A.: None End behavior: lim f ( x) = −∞ lim f ( x ) = ∞ f ( x) = x Square Root Function Domain: [0, ∞) Range: [0, ∞) Continuous Increasing: [0, ∞) Decreasing: None Symmetry: None Bounded below Extrema: Absolute minimum at (0,0) H.A.: None V.A.: None End behavior: lim f ( x) Does Not Exis t lim f ( x ) = ∞ f ( x) = ex f ( x ) = ln( x ) Natural Logarithm Function Domain: (0, ∞) Range: ( −∞, ∞ ) Continuous Increasing: (0, ∞) Decreasing: None Symmetry: None Not bounded Extrema: None H.A.: None V.A.: x = 0 End behavior: lim f ( x) Does Not Exis t lim f ( x ) = ∞ x→−∞ x f ( x ) = sin( x ) x→−∞ x→−∞ f ( x ) = x2 x →∞ f ( x) = 1 x x→∞ x→−∞ x→∞ f ( x ) = cos( x ) Cosine Function Domain: ( −∞, ∞ ) Range: [−1, 1] Continuous Increasing: [π + 2π n, 2π + 2π n], n ∈ Decreasing: [2 πn, π+ 2 πn], n∈  Symmetry: y-axis (even function) Bounded above and below Extrema: minimum at x = π + 2π n, n ∈  maximum at x = 2 π n, n ∈  End behavior: Oscilates between -1 and 1 f ( x ) = int( x ) Greatest Integer Function Domain: ( −∞, ∞ ) Range:  Not continuous Constant on the interval [ n, n + 1), n ∈  Symmetry: None Not bounded Extrema: None H.A.: None V.A.: None End behavior: lim f ( x) = −∞ lim f ( x ) = ∞ x→∞ x →−∞ →∞ Reciprocal Function Domain: ( −∞, 0) ∪ (0, ∞) Range: ( −∞, 0) ∪ (0, ∞) Not Continuous Increasing: None Decreasing: ( −∞, 0) ∪ (0, ∞) Symmetry: origin (odd function) Not bounded Extrema: None H.A.: y = 0 V.A.: x = 0 End behavior: lim f ( x) = 0 lim f ( x ) = 0 x x →−∞ →∞ Exponential Function Domain: ( −∞, ∞ ) Range: (0, ∞) Continuous Increasing: ( −∞, ∞ ) Decreasing: None Symmetry: None Bounded below Extrema: None H.A.: y = 0 V.A.: None End behavior: lim f ( x) = 0 lim f ( x ) = ∞ x x →∞ →−∞ Sine Function Domain: ( −∞, ∞ ) Range: [−1, 1] Continuous Increasing: −[ π + 2π n,π + 2π n] n∈ Decreasing: [ π 2 2 2 Symmetry: origin (odd function) Bounded above and below π Extrema: minimum at x = − + 2π n n ∈  2 π maximum at x = 2 + 2π n n ∈  End behavior: Oscilates between -1 and 1 →∞ x→−∞ Squaring Function Domain: ( −∞, ∞ ) Range: [0, ∞) Continuous Increasing: [0, ∞) Decreasing: ( −∞, 0] Symmetry: y-axis (even function) Bounded below Extrema: Absolute minimum at (0,0) H.A.: None V.A.: None End behavior: lim f ( x ) = ∞ lim f ( x ) = ∞ x f ( x) = x f (x ) = 1 1 + e−x Absolute Value Function Domain: ( −∞, ∞ ) Range: [0, ∞) Continuous Increasing: [0, ∞) Decreasing: ( −∞, 0] Symmetry: y-axis (even function) Bounded below Extrema: minimum at (0,0) H.A.: None V.A.: None End behavior: lim f ( x ) = ∞ x →∞ Logistic Function Domain: ( −∞, ∞ ) Range: (0, 1) Continuous Increasing: ( −∞, ∞ ) Decreasing: None Symmetry: None Bounded above and below Extrema: H.A.: y = 0 and y = 1 V.A.: None End behavior: lim f ( x) = 0 lim f ( x ) = 1 x →−∞ x→∞ + 2π n, 3π + 2π n] n ∈ 2