To Define a Function by a Symbolic Result

Symbolics > Symbolic Evaluation > To Define a Function by a Symbolic Result

To Define a Function by a Symbolic Result

1. Evaluate an expression symbolically.

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  <math resultRef="8">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <plus />
        <apply>
          <plus />
          <apply>
            <plus />
            <apply>
              <pow />
              <id xml:space="preserve">x</id>
              <real>2</real>
            </apply>
            <apply>
              <mult />
              <real>3</real>
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
          <apply>
            <mult />
            <real>2</real>
            <apply>
              <pow />
              <id xml:space="preserve">x</id>
              <real>2</real>
            </apply>
          </apply>
        </apply>
        <apply>
          <scale />
          <real>5</real>
          <id xml:space="preserve">x</id>
        </apply>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <plus />
          <apply>
            <mult />
            <real>3</real>
            <apply>
              <pow />
              <id labels="*" xml:space="preserve">x</id>
              <real>2</real>
            </apply>
          </apply>
          <apply>
            <mult />
            <real>8</real>
            <id labels="*" xml:space="preserve">x</id>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

2. Assign the symbolic expression to function f(x).

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  <math resultRef="9">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">f</id>
        <boundVars>
          <id labels="VARIABLE" xml:space="preserve">x</id>
        </boundVars>
      </function>
      <symEval>
        <apply>
          <plus />
          <apply>
            <plus />
            <apply>
              <plus />
              <apply>
                <pow />
                <id xml:space="preserve">x</id>
                <real>2</real>
              </apply>
              <apply>
                <mult />
                <real>3</real>
                <id xml:space="preserve">x</id>
              </apply>
            </apply>
            <apply>
              <mult />
              <real>2</real>
              <apply>
                <pow />
                <id xml:space="preserve">x</id>
                <real>2</real>
              </apply>
            </apply>
          </apply>
          <apply>
            <scale />
            <real>5</real>
            <id xml:space="preserve">x</id>
          </apply>
        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <mult />
              <real>3</real>
              <apply>
                <pow />
                <id labels="*" xml:space="preserve">x</id>
                <real>2</real>
              </apply>
            </apply>
            <apply>
              <mult />
              <real>8</real>
              <id labels="*" xml:space="preserve">x</id>
            </apply>
          </apply>
        </symResult>
      </symEval>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

3. Evaluate the function f(x) symbolically.

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  <math resultRef="10">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">f</id>
        <id xml:space="preserve">x</id>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <plus />
          <apply>
            <mult />
            <real>3</real>
            <apply>
              <pow />
              <id labels="*" xml:space="preserve">x</id>
              <real>2</real>
            </apply>
          </apply>
          <apply>
            <mult />
            <real>8</real>
            <id labels="*" xml:space="preserve">x</id>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

4. Evaluate the function f(x) numerically.

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Additional Information

When you define a function using the output of a symbolic evaluation, variables that you defined previously are ignored.

1. Evaluate an expression symbolically.

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  <math resultRef="12">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <minus />
        <apply>
          <pow />
          <id xml:space="preserve">y</id>
          <real>2</real>
        </apply>
        <real>5</real>
      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
        </sequence>
      </command>
      <symResult>
        <matrix rows="2" cols="1">
          <apply>
            <nthRoot />
            <placeholder />
            <real>5</real>
          </apply>
          <apply>
            <neg />
            <apply>
              <nthRoot />
              <placeholder />
              <real>5</real>
            </apply>
          </apply>
        </matrix>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

2. Assign a numerical value to y.

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3. Re-evaluate the expression symbolically.

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The evaluation fails because the input expression has no variable to solve for.

4. Assign the symbolic expression to function f(y), and evaluate it symbolically.

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  <math resultRef="15">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">f</id>
        <boundVars>
          <id labels="VARIABLE" xml:space="preserve">y</id>
        </boundVars>
      </function>
      <symEval>
        <apply>
          <minus />
          <apply>
            <pow />
            <id xml:space="preserve">y</id>
            <real>2</real>
          </apply>
          <real>5</real>
        </apply>
        <command>
          <sequence>
            <id xml:space="preserve">solve</id>
            <id xml:space="preserve">y</id>
          </sequence>
        </command>
        <symResult>
          <matrix rows="2" cols="1">
            <apply>
              <nthRoot />
              <placeholder />
              <real>5</real>
            </apply>
            <apply>
              <neg />
              <apply>
                <nthRoot />
                <placeholder />
                <real>5</real>
              </apply>
            </apply>
          </matrix>
        </symResult>
      </symEval>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

The evaluation passes because the assignment y:=2 is ignored.