Property 7
In functions defined from \(\mathbb{R}\) to \(\mathbb{R}\), in order to determine the result of a value, the desired value is substituted in place of the variable in the defined function.
Example
\(f(x)=x^2+3x-1\Rightarrow f(2)=?\)
Solution
\(f(2)=2^2+3\cdot2-1=9\)
1.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=x+3\)
\(\Rightarrow f(4)=?\)
2.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=x^2+5x\)
\(f(2)=?\)
3.
\(f:\mathbb{R}\setminus\{3\}\to \mathbb{R}\setminus\{2\}\)
\(f(x)=\samefrac{2x-4}{x-3}\)
\(\Rightarrow f(4)=?\)
4.
\(f:\mathbb{R}^+\to (-3,\infty)\)
\(f(x)=\sqrt{x}-3\)
\(\Rightarrow f(9)=?\)
5.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=x^3+3x^2+3x+1\)
\(\Rightarrow f(2)=?\)
6.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=3^{x-2}\)
\(\Rightarrow f(3)=?\)
7.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=x^2+kx+4\)
\(f(2)=8\)
\(\Rightarrow k=?\)
Property 8
Example
\(f:\mathbb{R}\to \mathbb{R}\)
\(f\left(\samefrac{x-2}{3}\right)=x^2+2x-7\Rightarrow f(1)=?\)
Solution
\(\samefrac{x-2}{3}=1\)
\(x-2=3\)
\(x=5\)
\(\to\text{in the function 5 is substituted by x}\)
\(f\left(\samefrac{5-2}{3}\right)=5^2+2\cdot5-7\)
\(f(1)=25+10-7\)
\(f(1)=28\)
1.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x-3)=x^2-1\)
\(\Rightarrow f(1)=?\)
2.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x+2)=3x-1\)
\(\Rightarrow f(4)=?\)
3.
\(f:\mathbb{R}\setminus\{3\}\to \mathbb{R}\setminus\{1\}\)
\(f(x+3)=\samefrac{x+1}{x}\)
\(\Rightarrow f(1)=?\)
4.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x-2)=x^2+x\)
\(\Rightarrow f(3)=?\)
5.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x^2+x)=x^2+x-6\)
\(\Rightarrow f(5)=?\)
6.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x^2+x)=2x^2+2x+8\)
\(\Rightarrow f(4)=?\)