Find values using function graphs

Learning Objectives Evaluate function values, f(x), for a given input x by reading its graph. Determine the input value(s) x for a given function output, f(x) = k. Identify the domain and range of a function using interval notation from its graph. Visually estimate the limit of a function at a point, including at points of discontinuity. Identify x-intercepts, y-intercepts, and intervals where the function is positive or negative. Locate local and absolute extrema (maxima and minima) on a function's graph. Determine the intervals where a function is increasing, decreasing, or constant. How can an engineer analyze the stress on a bridge over time without a single equation? 🌉 They read it from a graph! This tutorial focuses on the critical skill of interpreting functi...

TermDefinitionExample Function Value, f(c)The output or y-value of the function when the input is x = c. On a graph, it is the y-coordinate of the point (c, f(c)).For a graph of the parabola y = x², the function value f(3) is 9, found at the point (3, 9). Limit at a Point, lim_{x->c} f(x)The y-value that the function approaches as x gets infinitesimally close to c from both the left and the right sides. This value can exist even if f(c) is undefined or different.A graph has a hole at (2, 4) but a solid point at (2, 1). Here, f(2) = 1, but lim_{x->2} f(x) = 4. DomainThe set of all possible input values (x-values) for which the function is defined. Visually, it's the horizontal 'shadow' the graph casts on the x-axis.The graph of f(x) = sqrt(x-2) starts at x=2 and extend...

Finding a Function Value To find f(a), locate x=a on the x-axis. Move vertically (up or down) to the graph. Then, move horizontally to the y-axis to read the corresponding y-value. This is the fundamental process for evaluating a function at a specific point from its graph. Pay close attention to solid dots versus open circles. Solving for an Input Value To find the value(s) of x for which f(x) = k, draw a horizontal line y = k. The x-coordinate of every point where the line intersects the graph is a solution. Use this to work backwards from an output to find the corresponding input(s). A function can have multiple x-values for the same y-value. The Limit Existence Condition lim_{x->c} f(x) = L if and only if lim_{x->c^-} f(x) = L and lim_{x->c^+} f(x) = L...