HexMathv1.0.3

Function calculator with steps

Type or scan a function problem. HexMath evaluates, composes or solves it and shows each substitution.

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What this page covers

  • Evaluating f(x)f(x) at a number or at an expression such as x+hx + h.
  • Composing functions: f(g(x))f(g(x)) and g(f(x))g(f(x)).
  • Solving f(x)=kf(x) = k for xx, and simplifying the difference quotient f(x+h)−f(x)h\frac{f(x + h) - f(x)}{h}.
  • Not covered here: the domain and range (use Domain and range calculator) and the inverse of a function (use Inverse function calculator).

How to enter the problem

  • Type the definition first and then the question, for example f(x) = 2x^2 − 3x + 1, find f(−2).
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Let f(x)=x2+1\displaystyle f(x) = x^2 + 1 and g(x)=3x−2\displaystyle g(x) = 3x - 2. Find f(g(x))\displaystyle f(g(x)).

Answer

Verified

The answer is

f(g(x))=9x2−12x+5f(g(x)) = 9x^2 - 12x + 5

Explanation

  1. Put g(x)g(x) in place of the input of ff
    f(g(x))=(3x−2)2+1f(g(x)) = (3x - 2)^2 + 1
  2. Expand the square
    (3x−2)2=9x2−12x+4(3x - 2)^2 = 9x^2 - 12x + 4
  3. Add the constant
    9x2−12x+4+1=9x2−12x+59x^2 - 12x + 4 + 1 = 9x^2 - 12x + 5

Example 2

Problem

Let f(x)=x2−2x\displaystyle f(x) = x^2 - 2x. Solve f(x)=15\displaystyle f(x) = 15.

Answer

Verified

The answer is

x=5,x=−3x = 5, x = -3

Explanation

  1. Set the function equal to 15
    x2−2x=15x^2 - 2x = 15
  2. Move everything to one side
    x2−2x−15=0x^2 - 2x - 15 = 0
  3. Factor
    (x−5)(x+3)=0(x - 5)(x + 3) = 0
  4. Solve each factor and check
    x=5x = 5 or x=−3x = -3. Check: f(5)=25−10=15f(5) = 25 - 10 = 15 and f(−3)=9+6=15f(-3) = 9 + 6 = 15.

Example 3

Problem

For f(x)=x2−4x\displaystyle f(x) = x^2 - 4x, simplify f(x+h)−f(x)h\displaystyle \dfrac{f(x + h) - f(x)}{h}.

Answer

Verified

The answer is

2x+h−42x + h - 4

Explanation

  1. Evaluate f(x+h)f(x + h)
    (x+h)2−4(x+h)=x2+2xh+h2−4x−4h(x + h)^2 - 4(x + h) = x^2 + 2xh + h^2 - 4x - 4h
  2. Subtract f(x)f(x)
    x2+2xh+h2−4x−4h−(x2−4x)=2xh+h2−4hx^2 + 2xh + h^2 - 4x - 4h - (x^2 - 4x) = 2xh + h^2 - 4h
  3. Divide by hh
    2xh+h2−4hh=2x+h−4\frac{2xh + h^2 - 4h}{h} = 2x + h - 4, for h≠0h \neq 0.

Common mistakes

  • Squaring a negative input without brackets. f(−2)f(-2) needs 2(−2)2=82(-2)^2 = 8; reading it as 2⋅(−22)=−82 \cdot (-2^2) = -8 gives −1-1 instead of 15.
  • Reading f(g(x))f(g(x)) as the product f(x)⋅g(x)f(x) \cdot g(x). Composition puts g(x)g(x) inside ff.
  • Composing in the wrong order. g(f(x))=3x2+1g(f(x)) = 3x^2 + 1, a different function from f(g(x))f(g(x)).
  • Subtracting only the first term of f(x)f(x) in a difference quotient. The bracket in −(x2−4x)-(x^2 - 4x) changes both signs.

Checks, assumptions and limits

  • Test a composition at one number. With x=1x = 1: g(1)=1g(1) = 1 and f(1)=2f(1) = 2, and 9−12+5=29 - 12 + 5 = 2 too.
  • Functions here are real-valued; an input outside the domain has no output.
  • In a difference quotient hh is not zero, which is what allows dividing by hh.
  • HexMath can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

Evaluate it at a number, find a composition such as f(g(x)), solve f(x) = k, or simplify a difference quotient. Write the definition first, then the question.

Yes. Define f and g in the same problem, for example f(x) = x^2 + 1 and g(x) = 3x − 2, then ask for f(g(x)).

Yes. The first step writes the input in place of x, in brackets, before any arithmetic. That is where most sign errors start.

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Last updated: · HexMath